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Hydraulics with working tables,
3 1924 003 904 152
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HYDRAULICS
WITH
WORKING TABLES
BY
E. S. BELLASIS, M.Inst.C.E.
RECENTLY SUPERINTENDING ENGINEER IN THE IRRIGATION BRANCH
OF THE PUBLIC WORKS DEPARTMENT OF INDIA
AUTHOR OF 'RIVER AND CANAL ENGINEERING,* 'IRRIGATION WORKS,' ETC.
THIRD EDITION
LONDON
CHAPMAN & HALL, LTD.
1920
PEEFACE TO THII^D EDITION
In this edition the book has been brought thoroughly up to date
and subjected to careful and drastic revision. The chief object is,
as before, to deal thoroughly with the facts, laws, and principles of
Hydraulics, and to keep always in view their practical aspects.
The enormous waste caused by the use of erroneous co-efficients
is known to all. Another important object is to remedy this. The
use of old and inaccurate figures, — as some recent papers show — is
not uncommon. Fresh discussions on all the most important co-
efficients are now given and specific recommendations are made.
A new set of co-efficients for pipes is given.
Numerous examples of practical problems are included, and full
sets of tables for working them out.
The large quantity of new and original matter which — as some
reviewers were good enough to say — characterised the previous
editions is reproduced in an improved form, and fresh matter has
been added on weirs and weir-like conditions, on discharge measure-
ment by means of pipe diaphragms, on standing waves and the
practical use now being made of them in America, and on the laws
governing silting and scour. The difficult question of the surface
curve — ^upstream of weirs, etc. — is made clearer, and a simple
method of proceedure, applicable to a vast number of cases and
avoiding the use of the erroneous backwater function, is put
forward.
Some remarks are made on the best practical forms for the chief
formulee, and also some remarks — much needed — on the practice
of basing formulee on certain selected experiments while rejecting
others which are as good or better.
Obsolete matter and needlessly long mathematical investigations
are avoided.
It is hoped that the book will meet all the requirements both of
the student and of the engineer.
E. S. B.
Guildford,
IdS-ipt 1920.
CONTENTS
Chapter L— INTEODUCTION
PAGE
Section I. — Preliminary Remarks and Definitions —
Article 1. Hydravlics, 1
,, 2. Fluids, Streams, and Channels, .... 1
„ 3. Velocity and Discharge, ...... 2
Section II. — Phenomena observed in Flowing Water —
Article 4. Irregular Character of Motion, .... 3
,, 5. Contraction and Expansion, ..... 4
Section III.— Useful Figures-
Article 6. Weights and Measures, ...... 5
,, 7. Gravity and Air Pressure, . = .... 6
Section lY. — History and Remarks —
Article 8. Historical Summary, ... . . 7
,, 9. Remarks .8
Chapter II.— GENERAL PRINCIPLES AND FORMULiE
Section I. — First Principles —
Article 1. Bernouilli's Theorem, ... . 9
,, 2. Loss of Head from. Resistances, . . 11
„ 3. Atmospheric and other Pressures. . . 11
Section II. — Flow through Apertures —
Article 4. Definitions, .....■.,,. 12
,, 5. Flow through Orifices, . . » . . ., 13
,, 6. Flow over Weirs, ........ 15
,, 7. Concerning hoth Orifices and Weirs, , „ , . 16
Section III. — Flow in Channels —
Article 8. Definitions, . . ..... 19
9. Uniform Flow in Channels, ..... 20
10. Variable Flow in Channels, 22
11. Concerning both Uniform and Variable Flow, . . 24
12. Relative Velocities in Cross-section, . . 26
13. Bends, . 26
Section IV. — Concerning both Apertures and Channels —
Article 14. Comparisons of different Cases, 27
,, 15. Special Conditions affecting Flow, . . .29
,, 16. Remarks, .... . . 30
iV HYDRAULICS
PAOE
Section Y.— Abrupt and other Changes in a Channel-
Article 17. Abrupt Changes in General, 31
„ 18. Abrupt Enlargement, 31
,, 19. Abrupt Contraction • • ■ 33
20. Abrupt Bends, Bifii/rcations, and Junctions, . . 33
21. Concerning all Abrupt Changes, 34
Section ¥1.— Movement of Solids by a Stream-
Article 22. Definitions, 36
„ 23. General Laws 36
,, 24. Distribviion of Silt Charge .39
Section ¥11.— Hydraulic Observations and Co-efficients—
Article 25. Hydraulic Observations, 39
,, 26. Co-efficients,. ...... • . 41
Chapter III.— OEIFICES"
Section I.— Orifices in General-
Article 1. General Information, ... ... 43
, , 2. Measurement of Head, 44
„ 3. Incomplete Contraction, 45
„ 4. Changes in Temperature and Condition of Waier, . 47
,, 5. Velocity of Approach, 48
6. Effective Head, . . 50
„ 7. Jet from an Orifice, . ... .50
Section II.— Orifices in Thin Walls-
Article 8. Values of Co-efficient . . 53
„ 9. Co-efficients of Velocity and Oontraciion, ... 55
„ 10. Co-efficients for Submerged Orifices, .... 55
„ 11. Memarks, 55
Section III. — Short Tubes —
Article 12. Cylindrical Tubes, ... ... 57
,, 13. Specialformsof Cylindrical Tubes, .... 59
„ 14. Bell-mouthed Tubes, 61
„ 15. Conical Converging Tubes, 61
„ 16. jSTozzles, . . 62
17. Diverging Tubes, 64
Section lY.— Special Cases —
Article 18. Sluices and other Apertures, .... 69
,, 19. Vertical Orifices with Small Heads, . . .70
Examples, .... .... ... 72
Tables-
Table I. — Heads and Theoretical Velocities, . . 74
II — Imperfect and Partial Contraction, ... 76
III. — Co-efficients of Correction for Velocity of Approach, 76
IV. -VIII. — Co-efficients of Discharge for Orifices in Thin Walls, 77
v., IX. — Co-efficients of Discharge for Cylindrical Tubes, . 79
X. — Co-efficients of Correction for Vertical Orifices with
Small Heads 80
CONTENTS
Chapter IV.— WEIRS
Section I. — Weirs in General —
Article 1. General Information, ,
„ 2. Formulce,
,, 3. Incomplete Contraction,
„ 4. Flow of Approach,
,, 5. Velocity of Approach, .
81
84
85
85
86
Section II.— Weirs in Thin Walls-
Article 6. Co-efficients of Discharge, ...... 90
,, 7. Laws of Variation of Co-efficients, ■ ... 92
„ 8. Floiv when Air is excluded, , ... .93
9. Remarks, ...,,.... 93
Section III. — Other Weirs —
Article 10. Weirs with flat top and vertical face and hack, . . 96
„ 11. Weirs with sloping face or hack, . . .97
,, 12. Miscellaneous Weirs, . . .98
Section lY. — Submerged Weirs —
Article 13. Weirs in Thin Walls, ... . .99
„ 14. Other Weirs, ........ 103
,, 15. Contracted Channels and Weir-like Conditions, . . 105
Section V.— Special Cases-
Article 16. Weirs icith Sloping or Stepped Side-walls, . . . Ill
„ 17. Canal Notches, 112
„ 18. Oblique and Special Weirs 116
Examples, 118
Tables-
Table -x.!. — Values of Hand Hi 120
:k.ii.— Values of K or %c J^ (/r h-Zhc, . . .121
XIII. — Co-efficients of Correction for Velocity of Ap-
proach 121
XIV. -XVI. — Co-efficients of Discharge for Weirs in Thin
Walls, 122
XYii. — Corrections for Wide Crests, . . . 123
xvili.-xxii. — Inclusive Co-efficients for Weirs, . . . 124
Chapter V.— PIPES
Section I. — Unifonn Flow —
Article
1. General Information,
. 127
2. Short Pipes, , . ....
. 130
3. Combinations of Pipes
. 130
4. Bends,
. 133
5. Belative Velocities in Cross-section,
. 135
VI HYDRAULICS
PAGE
Section n. — Variable Flow-
Article 6. Abrupt Changes, 136
, , 7. Gradual Changes, 139
Section HI. — Co-efBcients and Fonuulse —
Article 8. General Information, ... . , . . .142
„ 9. Co- efficients for Ordinary Clean Pipes, .... 143
„ 10. Co-efficients for Other Pipes, 148
„ 11. Formnlai 152
Notes to Chaptek V, ... .... 154
Examples, . . 155
Tables-
Table XXIII. — Values of A and R for Circular Sections, . 158
,, xxiv.-xxvB. — Co-effixients for Pipes, 159
Notes on Hydraulic Tables • . . 160
Table xxvi, — Values of C iJM for various Values of C and
JB 163
,, XXVII. — Values of S and ^S, 167
„ xxvili. — Velocities for various Values of JS and C JR, 168
Chapter VI.— OPEN CHANNELS— UNIFORM FLOW
Section I. — Open Channels in General —
Article 1. General Information, ....... 172
„ 2. Laws of Variation of Velocity and Discharge . . 173
Section n. — Special Forms of Channel-
Article 3. Section of ' Best Form,' . ... . . 175
,, 4. Irregular Sections, ..... . 177
„ 5. Channels of Constant Velocity or Discharge, . . , 178
,, 6. Circular Sections, . . . . . . . 17Q
Section IIL — Relative Velocities in Cioss-section —
Article 7. General Laws, ........ 179
,, 8. Horizontal Velocity Curves, ...... 181
,, 9. Vertical Velocity Curves, 184
„ 10. Central Surface Velocity Co-efficients, .... 1S9
Section IV. — Co-eiiicients —
Article 11. Bazin's and Kutter's Co-effi/iietUs, 190
,, 12. Rugosity Co-efficients ....... 194
„ 13. Remarks, ......... 196
Section V.— Movement of Solids by a Stream-
Article 14. FormtUai and their Application I97
,, 15, Remarks, ..... . . 20'2
Notes to Chaptbe VI \ _ 205
Examples, ... 206
Tables-
Table xxix.-XLil.— ^Mttcr's, Bazin's, and Manning's Co-ifficients
with C ^R 209
„ xLiii.-XLVi. — Tatees of A and ,fR for Open Channels {Rect-
angular, J <o 1, 1 to 1, 14 to 1), , . . 223
, , XLvii. -XLix. — Values of A and JRfor Oval Sections {Sewers), 245
)) !■ — For calculating Lengths of Side-slopes, . . 248
11 LA. — Sectional Data for Circular Sections, , . 248
CONTENTS VJl
Chapter VIL— OPEN CHANNELS— VARIABLE FLOW
PACE
Section I. — Bends and Abrupt Changes —
Article 1. Bends, . . ...... 249
„ 2. Changes of Section, . 251
„ 3. Bifurcations and Junctions, . . . 253
, , 4. Relative Velocities in Cross-section, . . . 254
Section II.— Yariable Flow in a Uniform Channel {General De-
scription) —
Article 5. Breaks in Uniformity, .... . . 254
,, 6. Bifurcations and Junctions, . . . 258
,, 7. Effect of Change in the Discharge, .... 269
,, 8. Effect of Alterations in a Channel, . . , 260
„ 9. Effect of a Weir or Raised Bed 262
Section III. — Variable Flow in a Uniform Channel (Formula! and
Analysis) —
Article 10. Formulae, ....,, . . 264
„ 11. Standing Wave, . . . . , . . 266
,, 12. The Surface-curve, .... 271
,, 13. Metliod of finding Surface-curve, . . . . 272
„ 14. Calculations of Discharges and Water-levels, . . 280
Section lY. — Yariable Flow in General —
Article 15. Flow in a Variable Channel, .... 281
,, '16. Uniform and Variable Flow, 283
„ 17. Rivers, 284
Notes to Ohaptkr VII., ........ 284
Examples, . . ... 2S6
Tables-
Table LI. — Ratios for calculating Profile of Surface when headed
up,. . .
, , Lii. — Ratios for calculating Profile of Surface when drawn
down, . ...... 289
288
Chapter VIII.— HYDRAULIC OBSERVATIONS
Section I.— General —
Article 1. Velocities, ... . . .... 290
,, 2. Discharges, ......... 292
„ 3. Soundings, . , i . , . . _ . 295
4. Miscellaneous, . . , . . , , . 295
Section n. — ^Water-levels and Pressure Heads —
Article 5. Gauges, . ^ ...... . 297
6. Piezometers, .... ... 299
7. Surface-slope, . 300
Section III.— Floats —
Article 8. Floats in General, . = , . 301
,, 9. Sub-surface Floats, . . , , . , . 303
„ 10. Rod Floats, . ........ 306
VIU HYDRAULICS
FAOE
Section lY. — Current-meters —
Article 11. General Description, ... • • 308
,, 12. Varieties of Gurrent-metem, , ■ • 310
„ 13. Bating of Current-meters, ... . • 315
Section Y Pressure Instruments-
Article 14. Pitot's Tube 319
,, 15. Other Pressii/re Instruments, 321
Section VI.— Pipes-
Article 16. The Venturi Meter, '^ . 321
„ 17. Pipe Diaphragms, 322
Notes to Chapter VIII 324
Chapter IX.— UNSTEADY FLOW
Section I. — Flow from Orifices —
Article 1. Head uniformly varying, . . . 326
,, 2. Filling or Em/ptying of Vessels, , , . . 326
Section II. — Flow in Open Channels —
Article 3. Simple Waves, ...... . 328
,, 4. Complex Cases, . , > . . • • 331
„ 5. Remarks, . . . ^ 332
* Chapter X.— DYNAMIC EFFECT OF FLOWING'WATER
Section I, — General Information —
Article 1. Preliminary Remarks, ...... 334
„ 2. Radiating and Circular Currents, , . , 334
Section II. — Reaction and Impact —
Article 3. Reaction, .... ■ . ■ . 336
„ 4. Impact, , 337
„ 5. Miscellaneous Causes, . , , . . 341
APPENDICES
Appendix A. — Units, 343
,, ^, —Calculation of m and n ...... 343
,, C. — Formulae, _ _ 344
,', B. — Variable Flow, 345
,, E. — Unsteady Flow 345
INDBX, g^g
CHAPTER I
INTRODUCTION
Section I. — Preliminary Eemarks and Definitions
1. Hydraulics. — Hydraulics is the science in which the flow
of water, occurring under the conditions ordinarily met with in
Engineering practice, is dealt with. Based on the exact sciences
of hydrostatics and dynamics, it is itself a practical, not an exact,
science. Its principal laws are founded on theory, but owing
to imperfections in theoretical knowledge, the algebraic formulsa
employed to embody these laws are somewhat imperfect and con-
tain elements which are empirical, that is, derived from observation
and not from theory. The science of Hydraulics is concerned with
the discussion of laws, principles, and formulae, of such observed
phenomena as are connected with them, and of their practical
application. The quantities dealt with are generally velocities
and discharges, but sometimes they are pressures or energies. It
is frequently necessary in Hydraulics to refer to particular works
or machines, but this is done to afford practical illustrations of
the application of the laws and principles. Descriptions of works
or machines form part of Hydraulic Engineering and not of
Hydraulics, and the same remark applies to statistical information
on subjects such as Eainfall. Some description of Hydraulic
Fieldwork is included in this work for reasons given below
(chap. ii. art. 25). The laws governing the power of a stream to
move solids by rolling or carrying them are intimately connected,
with the laws of flow and are naturally included.
2. Fluids, Streams, and Channels. — A 'fluid' is a substance
which offers no resistance to distortion or change of form. Fluids
are divided into ' compressible fluids ' or ' gases/ such as air, and
' incompressible fluids ' or ' liquids,' such as water. Perfect fluids
are not met with, all being more or less 'viscous,' that is, offering
some resistance, though it may be very small, to change of
form. A ' stream ' is a mass of fluid having a general movement of
A
HYDRAULICS
Fig. 1.
translation. It is generally bounded laterally by solid substances
which form its 'channel.' If the channel completely encloses the
stream, and is in contact with it all round, as in a pipe running
full, it is called a ' closed channel ' ; but if the upper surface of the
stream is 'free,' as in a river or in a pipe running partly full, it is
an ' open channel.' An ' eddy ' is a portion of fluid whose particles
have movements which are irregular and generally more or less
rotatory ; it may be either stationary or moving with respect to
other objects. The ' axis ' of a stream or channel is a line centrally
situated and parallel to the direction of flow. In
an open channel its exact position need not be fixed,
but in a pipe it is supposed to pass through the
centre of gravity of each cross-section.
An 'orifice' or 'short tube' (Fig. 1) is a short
closed channel expanding abruptly, or at least very
rapidly, at both its upstream and downstream ends.
A short open channel similarly circumstanced
(Fig. 2) is called a ' weir,' provided the expansions
are wholly or partly in a vertical direction. When they are
wholly lateral it is called a 'contracted channel.' All these short
channels will collectively be
termed 'apertures,' and 'channel'
will be used for channels of con-
siderable length.
The stream issuing from an ori-
fice or pipe is called a 'jet,' that
falling from a weir a ' sheet.' Except in the case of a jet issuing
under water a stream bounded by other fluid of the same kind is
called a 'current.'
3. Velocity and Discharge. — The direction of the flow of a
stream is in general parallel to the axis, but it is not always so at
each individual point. If at any point the flow is not parallel to
the axis, the velocity at that point may be resolved into two com-
ponents, one of which is parallel to the axis and the other at rio-ht
angles to it. The component parallel to the axis is termed the
'forward velocity.' A ' cross-section ' of a stream is a section at
right angles to the axis. The velocities at all points in the cross-
section of a stream are not equal. A curve whose abscissas
represent distances along a line in the plane of the cross-section,
and whose ordinates represent forward velocities, is called a
•velocity curve.' The 'discharge' of a stream at any cross-
section is the volume of water passing the cross-section in the
Fio. 2.
INTRODUCTION 3
unit of time. The ' mean velocity \ at any cross-section is the
mean of the different forward velocities. It is the discharge of
the stream divided by the area of the cross-section. Thus
F='^ovQ=AF . . . (1).
This is the first elementary formula of Hydraulics. Except when
velocities at individual points are under consideration, the term
'velocity' is generally used instead of 'mean velocity.'
As long as the conditions under which flow takes place at any
given cross-section of a stream remain constant, the velocity and
discharge are constant, that is, they are the same in succeeding
equal intervals of time. In this case the flow is said to be
'steady.' As soon as the conditions change, the velocity and dis-
charge usually change, and the flow is then said to be unsteady.
Owing to the introduction or abstraction of water by subsidiary
channels, leakage, or evaporation, the discharges at successive
cross-sections of a stream may be unequal, but the flow may still
be steady. Flow is unsteady only when the discharge varies
with the time, and not when it merely varies with the place. In
Hydraulics, flow is always assumed to be steady unless the con-
trary is expressly stated. For instance, in the statement that
a rise of surface level gives an increase in velocity, it must be
understood that this refers to the period after the surface has
risen, and not to that while it is rising. In any length of stream
in which the flow is steady, and in which no water is lost or
gained, the discharges at all cross-sections are equal, or
Q=J,F,=J,F, = etc., ... (2)
where A^, A,, etc., are the areas of the cross-sections, and Fi, F^,
etc., the mean velocities. In other words, the mean velocity at
any cross-section is inversely as the sectional area.
Section II. — Phenomena observed in Flowing Water
4. Irregular Character of Motion. — In flowing water the free
surface oscillates, especially in large and rapid streams. The
oscillation is probably greater near the sides than at the centre.
The motion of the water is also irregular. Except under peculiar
conditions the fluid particles do not move in parallel lines, or
'stream-lines,' but their paths continually cross each other, and
the velocity and direction of motion at any point vary every
instant. The stream is, in fact, a mass of small eddies. The
4 HYDRAULICS
irregularities of motion increase with the roughness of the channel
and with the velocity of the stream. They are especially great
in open channels. Eddies produced at the bed are constantly
rising to the surface. Floats dropped in at one point in quick
succession move neither along the same paths nor with the same
velocities. In experiments made by Francis,'whitewash discharged
into a stream four inches above the bed came to the surface in
a length which was equal to ten to thirty times the depth, and
was less, the rougher the channel. The eddies are strongest
where they originate, namely, at the border of the stream. To
compensate for the upward eddies there must, of course, be down-
ward currents, but they are diffused and hardly noticeable. The
resistance to flow caused by all these irregular movements is enor-
mously greater than that which would exist in stream-line motion.
Although the velocity and water-level at any point fluctuate
every moment as above described, the average values obtained
in successive periods of time of longer duration are more or less
constant. The velocities obtained at any point in successive
seconds will, perhaps, vary by 20 per cent. ; those obtained in
successive minutes will vary much less ; and those in successive
periods of five minutes each probably scarcely at all. The same
is true of the direction of the flow. For the water-level the
averages of several observations obtained in periods of a minute
each will probably agree very closely. A velocity curve obtained
from a few observations is generally irregular, but one obtained
from a large number is regular. If the flow is not steady, the
average velocities and water-levels obtained in successive long
periods of time may, of course, vary, but they will exhibit a
regular change. When velocity and water-level are spoken of,
the average values and not the momentary values are meant, and
this remark applies to the foregoing definition of steady flow.
The discharge at any cross-section, if considered in its momentary
aspect, is probably never steady. The irregularity of the motion
of water renders the theoretical investigation of flow extremely
difficult, and no complete theory has yet been propounded.
5. Contraction and Expansion. — Except under an infinite force,
a body cannot, without either coming to rest or describing a
curve, change its direction of motion. Acting in obedience to
this law, water cannot turn sharp round a corner. "Wherever any
sharp salient angle A or B (Fig. 3) occurs in a channel, or at the
entrance of an aperture, the water travelling along the lines GA,
SB cannot turn suddenly and follow the lines AC, BD. It follows
INTRODUCTION
D
s
b/
if
1
l^.^'
M
c
Pig. 3.
the lines ^^, BF, which are curves. At A and B the r£(!dii of the
curves may be very small, but the curves doubtless touch the lines
GA, HB. This phenomenon
is known as 'contraction.'
The stream contracts from
AB to BF. If the channel or
aperture extends far enough,
the stream expands again
and fills it at MN, the spaces
AME, BNF containing
eddies. These have, however,
little or no forward move-
ment, and are not part of
the stream. There are also
eddies at K, L. In a case of abrupt enlargement (Fig. 4) the
stream expands gradually, and
there are eddies in the corners.
Similar phenomena occur at
abrupt bends, bifurcations,
and junctions. For a closed
channel or an orifice. Fig. 3
represents any longitudinal
section. For an open channel
or a weir, it represents a plan
or a horizontal section, and its lower part — from PQR downwards
— a vertical section. And similarly with Fig. 4. Sometimes still
or ' dead ' water may replace part of an eddy. The term eddy
will be used to include it.
B'E
F'F
Fig. 4.
Section III.— Useful Figures
6. Weights and Measures. — The following table ^ gives the
weight of distilled water for various temperatures. The weights
of clear river and spring water are practically the same as the
above. For all ordinary practical purposes the weight of fresh
water may be taken to be 62-4 lbs. per cubic foot when clear,
and 62-5 lbs. or 1000 ounces when containing sediment. Water
is compressed by about one twenty-thousandth part of its bulk
by a pressure of one atmosphere. Sea-water weighs about 64 lbs.
per cubic foot. Water usually contains a small quantity of air in
solution.
^ Smith's Hydraulics, chap. i.
HYDRAULICS
Temperature
Pounds prjr
Temperature
Pounds per
Temperature
Pounds per
(Fahrenheit.)
Cubic Foot.
(Fahrenheit),'
Cubic Foot.
(Fahrenheit).
Cubic Foot.
32°
62-42
95°
62-06
160°
61 01
35°
62-42
100°
62-00
105°
60-90
39-3°
62-424
105°
61-93
170°
60-80
45°
62-42
110°
61-86
175°
60-69
50°
62-41
115°
61-79
180°
60-59
55°
62-39
120°
61-72
185°
60-48
60°
62-37
125°
61-64
190°
60-36
65°
62-34
130°
61-55
195°
60-25
70°
62-30
135°
61-47
200°
60-14
75°
62-26
140°
61-39
205°
60-02
80°
62-22
145°
61-30
210°
59-89
85°
62-17
150°
61-20
212°
59-84
90°
62-12
155°
61-11
An Imperial gallon of water contains ^^^ cubic feet, and weighs
almost exactly 10 lbs. A United States gallon is five-sixths of an
Imperial gallon. A metre is 3-2809 feet, a cubic metre 35-317
cubic feet, a kilogram 2-2055 pounds avoirdupois, and a litre
61-027 cubic inches or '2201 gallons. A cubic metre of water
weighs 1000 kilograms. The metric system being that chiefly
employed on the continent of Europe, these figures may be useful
in the conversion of figures given in reports of foreign experiments
or investigations. A French inch is -02707 of a metre or -0888 of
an English foot.
The units employed in this work are the foot, the second, and
the pound. Thus velocities and discharges are in feet or cubic
feet per second, weights in pounds per cubic foot.
7. Gravity and Air Pressure. — The force of gravity, denoted
by g, is generally assumed to be 32-2, that is, it is supposed to
increase the velocity of a falling body by 32-2 feet per second, and
J^g, a quantity very frequently occurring in hydraulics, is then
8-025. These figures are suitable for Great Britain and Canada,
but the force of gravity varies with the locality, increasing with
the latitude and decreasing with the height above sea-level. At
the Equator at the sea-level g is 32-09, and at the Pole at the
sea-level it is 32-26. The mean values of g and J^g for ordinary
elevations and for latitudes up to 70° are 3216 and 8-02 respec-
tively. These are suitable for the United States, India, and
Australia, and are adopted in this work. They, however, differ by
only -12 per cent, and -06 per cent, respectively from the values
given above, and ordinarily this difference is of no account what-
ever. An increase of elevation of 5000 feet decreases g by only
-016 and ^J2g by -002.
INTRODUCTION I
The pressure of the atmosphere near the sea-level is about 14'7
lbs. per square inch, and is equivalent to about 30 inches of
mercury or 34 feet of water. According to the 'English system'
of computation by 'atmospheres,' one atmosphere is equivalent
to 29"905 inches of mercury in London at a temperature of 32°
Fahrenheit. The French system gives a pressure which is greater
in the ratio of 1 to -QDOT. For elevations above the sea-level the
atmospheric pressure decreases. Up to a height of 6000 feet the
reduction for every thousand feet is about '5 lb. per square inch,
or 1 inch of mercury, or 1'13 feet of water. Above 6000 feet the
reduction is less rapid, amounting to 1'9 lbs. per square inch in
rising from 6000 to 11,000 feet.
Section IV. — History and Eemarks
8. Historical Summary. — A historical sketch of Hydraulics given
in the Encyclopmdia Britannica ^ comprises the names of Castelli,
Torrioelli, Pascal, Mariotte, Newton, Pitot, Bernouilli, D'Alembert,
Dubuat, Bossut, Prony, Eytelwein, Mallet, Vici, Hachette, and
Bidone. To these may be added Michelotti, D'Aubuisson, Castel,
and Borda.
Coming to specific branches of Hydraulics and recent periods,
flow in pipes has been made the subject of experiment and investi-
gation by Weisbach, Coulomb, Venturi, Couplet, Darcy, Lampe,
Hagen, Poiseuille, Reynolds, Smith, and Stearns, and flow through
apertures by Poncelet, Lesbros, Weisbach, Eennie, Blackwell,
Boileau, Ellis, Bornemann, Thompson, Francis, ^ Unwin, Fteley and
Stearns,^ Herschel, Steokel, Fanning, and Smith.* Many of the
experiments on pipes and apertures have been discussed and sum-
marised by Fanning ^ and Smith,* both of whom have compiled
tables of co-efBcients for pipes and apertures. Since then further
important experiments have been made on weirs by Bazin,'' on
weirs and pipes by various American engineers,' and on orifices,
weirs, and pipes by others who are mentioned in chapters iii.
to V.
^ Encydopcedia Britannica. 9th Edition. Article 'Hydromechanics.'
^ Lowell Hydraulic Experiments.
^ Transactions of the American Society of Civil Engineers, vol. xii.
* Hydraulics,
^ Treatise on Water Supply Engineering.
'' Annales des Pouts et Gliaussees. 6th Series, Tomes 16 and 19, and 7th
Series, Tomes 2, 7, 12, and 15. A risumi is given in Ultcoulement en
Deversoir.
7 Transactions of the American Society of Civil Engineers, vols, xix.,
8 HYDRAULICS
Eegarding flow in open channels, extensive observations and
investigations have been made by Darcy and Bazin ^ on small
channels, by Humphreys and Abbott ^ on the Mississippi, and by
Cunningham ^ on large canals. Many observations have also been
made by German engineers and some by Eevy* on the great
South American rivers. In this branch of Hydraulics the Swiss
engineers Ganguillet and Kutter have analysed most of the chief
experiments,* including some made by themselves, and arrived at
a series of co-eificients for mean velocity. Their writings have
been translated and commented on by Jackson," who has framed
tables of co-efficients ^ based on their researches. Finally Bazin
has reviewed the whole subject* and arrived at some fresh co-
efficients. Investigations have been made by Francis^ on rod-
floats, by Stearns i" on current-meters, and by Kennedy " on the
silt-transporting power of streams.
9. Remarks.— The different hranches of Hydraulics are shown
by the headings of chapters iii. to x. of this work. In the
following chapter the whole subject is considered in a general
manner. This enables us to dispose once for all of many points
which would otherwise have had to be mentioned in more than
one of the subsequent chapters. Moreover, the different branches
are not always divided by such hard and fast lines as might appear ;
there are many points common to two branches, and the preliminary
consideration of the various branches of the subject in connection
with one another instead of separately will be advantageous.
xxii., xxvi., xxviii., xxxii., xxxiv., xxxv., xxxvi., xxxviii., xl.,
xli., xlii., xliv., xlvi., xlvii.
' Recherches HydrauKques.
^ Report ore the Physics and HydravXica of the Mississippi Sire?:
' Rooriee Hydraulic Experiments-
* Hydraulics of Qreat Rivers.
<> A General Formula for the Uniform Flow of Water in. Rivers
and other Channels. Translated by Hering and Trautwine.
" The New Fonmda for Mean Velocity in Rivers and GancUs.
Translated by Jackson. For other writers see chap. vi.
' Canal and Culvert Tables.
^ Miided'nne Nouvelle Formule pour Canattx Decouncrts.
" Lowell Hydraulic Experiments.
'" Transactions of the American Society of Cii'i/ Engineers, vol. xii.
1' Minutes of Proceedings, Institution of Cieil Engineers, vol. oxix.
CHAPTER II
GENERAL PRINCIPLES AND FORMULAS
Section I. — First Principles
1. Bernouilli's Theorem. — Let Fig. 5 represent a body of still
water, the openings at F and F being supposed to be closed. The
Fig. 5.
water in the tubes at C, D stands at the same level as AB. The
' head ' or ' hydrostatic head ' over any point is its depth below the
plane AB. This plane is sometimes called the 'plane of charge.'
The pressure is as the head. If P is the pressure per square foot
at the depth 11, and W the weight of one cubic foot of water, then
P=WH or H—^fj-^ The head H is said to be that 'due to' the
}V
pressure P.
Every particle of water in the reservoir possesses the same degree
of potential energy. Comparing a particle at the depth H with one
at the surface, the one possesses energy in virtue of its pressure,,
the other in virtue of its elevation.
Let an orifice be opened at F so that water flows along the pipe
GEF, and let the reservoir be large, so that the water in it has no
velocity and the surface AB is unaltered. The pressure in the
water Sowing .in the pipe is reduced, and the water-levels in the
10 HYDRAULICS
tubes fall to K, L. The heights KM, LN are as the pressures at M
and N, and they are called the ' hydraulic heads ' or ' pressure heads.'
The tubes are called ' pressure columns,' and the line BKL the line
of ' hydraulic gradient.' Let p be the pressure at M, and hp the
pressure head. Then hp=^. Let F"be the velocity in the pipe at
ilf and let A^= — . Then A„ is the ' velocity head.' It is the height
through which a body falls under the influence of gravity in an
unresisting medium in acquiring the velocity V, or the height to
which it could be made to rise by parting with its velocity. Let
it be supposed that there are no resistances to the motion of the
water, so that no energy is consumed in overcoming them. Then
by the law of the conservation of energy the total energy of any
moving particle of water remains as before. Whatever is lost as
pressure is gained as velocity. The head ck lost in pressure is the
velocity head h^. Thus
h=hp+K ■ ■ ■ (3),
or the pressure head added to the velocity head is the hydrostatic
head. This equation, due to Bernouilli, is the basis of all theo-
retical hydraulic formulae. It obviously applies to any point in
the pipe.
It has been seen that the pressure at M is as the height K3L
Assume that the velocities at all points in the cross-section 2IQ are
equal. Let Up and H^ be the pressure head and velocity head
at E, then H= Hp+H^; h=hp+ h^.
But since the velocities are equal, R^=hy; therefore Hp—hp=
H—h, or the change in pressure in passing from M to E is the same
as it was when there was no flow. The pressure head at E is KE, and
the pressure at any point in the cross-section is as its depth below A'.
Let OP be a datum-line and let h^ be the ' head of elevation ' of
any point M above OP. Then h+h^ is constant for all points in the
system, and therefore
hp+K+h,=K ... (4)
where K is constant. This is Bernouilli's theorem more fully
stated. The total energy possessed by a particle of water is the
. sum of the energies due to its pressure, \eIocity, and ele\ation.
If instead of a pipe wn consider an open channel AT, the results
obtained will be the same as before. If pressure columns -were
used the water in them would not rise aljove the surface AT. At
each point in the surface tlu^ pressure head is zero and the velocity
GENERAL PRINCIPLES AND FORMULA 11
head is equal to the hydrostatic head. If the velocities at all
points in a cross-section are assumed to be equal, the law of change
of pressure with depth is the same as before.
Since the area NR is greater than MQ, the velocity is less and
the pressure greater. Thus from K to L there is a rise in the
hydraulic gradient. Similarly, in the open stream there is a rise
where the sectional area is increasing.
The pressure in a body of flowing water can never be negative,
as the continuity of the liquid would be broken.
2. Loss of Head from Resistances. — Practically a certain amount
of head h' is always expended in overcoming resistances, due to
the friction of the water on its channel and to the internal move-
ments of the water, so that the total head diminishes in going along
the stream in the direction of the flow. In other words, the
pressure head and velocity head do not together equal the hydro-
static head. The difference is the 'head lost.' The actual water-
levels would in practice be S, T, and CS, DT would be the total
losses of pressure head up to the points M and N. As head is
lost, the work which the water is capable of doing in virtue of its
elevation, pressure, and velocity is diminished. If h' is the head
lost by resistance between two cross-sections, then
^^=^'-^^ ■ ■ ■ (5).
or the head lost is equal to the fall in the surface or line of gradient
less the increase in the velocity head. The same is true of the
open channel. The surface would be XZ instead of XY.
3. Atmosplieric and other Pressures. — Generally a body of
water is subjected to the atmospheric pressure P„. The head
. p
due to this pressure is -.°, and this has to be added in order to
obtain the total head over any point. The case is the same as if
the water-surface at each point were raised from AD to UW by a
W
the relative heads over two or more points are considered, the
pressure of the atmosphere affects all parts equally and is left out
of consideration. If, however, different portions of the water are
subjected to pressures of different intensities caused, say, hj partly
exhausted air, by steam, or by a weighted piston, the water-surface
of each portion of the system must be considered as being raised
P
by a height -^^ where P is the intensity of the special pressure
acting on it.
height ^. But usually — as in the preceding demonstrations —
12
HYDRAULICS
Fig. 6.
Fig. 7.
Section II. — Flow through Apertures.
4. Definitions. — An aperture is said to be ' in a thin wall ' when
its upstream edge is sharp (Figs. 6 and 7), and the ' wall ' or structure
containing the aperture is thin, or is
bevelled or stepped, so that the stream
after passing the edge springs clear
and does not touch it again. An
aperture like that shown in Fig. 1 or
Fig. 2, page 2, may have its upstream
edge sharp, but it does not come
within the definition, i A rounded or
'bell-mouthed' orifice (Fig. 8) is one
in which the sides are curved, so that
the tangents at c, D are parallel, and the stream after passing CD
does not contract. A weir of analogous shape may be formed by
rounding the angle between the top and the upstream
side or 'face,' and by prolonging the sidewalls AB
(Fig. 8a) upstream.
The upstream surface of the wall surrounding an
aperture will be called the 'margin.' The margin is
said to be 'clear' when it is free from projections,
leakages, or anything which would interfere with
the free flow of water along the wall towards the
aperture. The clear margin, if not otherwise limited,
is bounded by the sides of the reservoir or channel,
or by any other aperture existing in the same wall.
When an aperture has sharp edges an increase in the
clear margin, up to a certain limit, increases the
degree of contraction. When this limit has been
reached the contraction is said to be 'complete.'
Fio. 8.
t-z;x
Tzzm
V777777777r?
ELEVATION
#=*
I
I
/ /
t
^m
PLAN.
Fig. 8a.
' Por this reiisou the oxproaaion ' sharp-edged,' used by some recent writers
in preference to the old one of ' in a thin wall,' is not suitable.
GENERAL PRINCIPLES AND FORMULA 13
5. Flow through Orifices. — Let H be the height of the free
surface (Fig. 9) above the centre of gravity of the small orifice
C, D, or E, and lot V be the velocity of the issuing jet. Both
the jet and the free surface AB are supposed to be subject to
the atmospheric pressure P„. The total head over the orifice
p
is S+j^., and the pressure in and upon the issuing jet is P„
Then' from equation 3 (page 10), supposing no head to be lost in
overcoming resistances,
or V= 4=igE . . . (6).
All formulse for flow from apertures are modifications of this.
The velocity \/2(/if is called the 'theoretical velocity.' It is the
same as would be acquired by a body falling from .rest in a vacuum
through a height H. If the jet issues vertically upwards it will,
in the absence of all resistance except gravity, rise to the level of
AB. The velocity depends only on II and not on the direction in
which the jet issues. If AGR is a parabola with axis vertical and
parameter 2(7, the theoretical velocities of jets issuing at F, M, N
are as the ordinates FG, MK, NR. Practically owing to resistances
caused by friction and internal movements of the water, the
velocity of efflux is less than the theoretical velocity, and is
given by the formula
V=c,J^H_ . . (7),
where c^ is a ' co-efficient of velocity ' whose mean value for the
two kinds of orifices under consideration is about '97.
Instead of assuming the water in the reservoir to have no
appreciable motion, let it be supposed that it is moving with a
velocity v directly towards the orifice. This velocity is called
' velocity of approach ' and the discharge through the orifice is in-
creased. The energy possessed by the water can, theoretically,
V
■ raise it to a height — or h. This is called the head due to the
2<7
velocity of approach, and it must be added to the hydrostatic head.
Practically, for reasons which will be given below, a head nh has
to be added, n being 1 -0 or less. The formula thus becomes
r^c,J-2g{II+nk) . . . (8).
If the fluid moved without resistance, a velocity v in any direction,
and not only toward the orifice, could be utilised in increasing the
14
HYDRAULICS
head and the discharge, but practically the only useful component
of the velocity is that parallel to the axis of the orifice.
In the case of an orifice in a thin wall (Fig. 6), the jet attains
a minimum cross-section at JB, whose distance from the edge of
the orifice is about half the diameter of the orifice, or half the least
diameter if the orifice is of elongated form. This minimum
section is called the ' vena oontracta.' The ratio of its sectional area
a' to the area a of the orifice is called the ' co-efficient of contrac-
tion,' and is denoted by c„: thus al — cfi,. The mean value of c^ is
about '63. A vena contracta occurs with any kind of orifice
having sharp edges, and c„ is probably about the same. For a
bell-mouth c„=l-0.
The discharge of an orifice is
Q = a! F= atX'v V 'igH..
Let cfi^ — c. Then c is the ' co-efiicient of discharge ' and
Q=acsJigB . . . (9).
Or when there is velocity of approach
Q=ac>Jlg{H+nh) . . . (10).
The value of c for orifices in thin walls averages about '61, and
for bell-mouthed orifices "97. It does not usually vary much with
the head. Generally the values of c„, c„, and c are not very greatly
afiected by the shape and size of an orifice nor by the amount of
head. Generally c is better known than c„ or c^, and it is also of
far more importance.
When an orifice has a head of water on both sides it is said to
be ' submerged ' or ' drowned,' and H in the formula is the differ-
ence between the two
■^ -B heads. Thus for any ori-
fice Q or (Fig. 9), the
head is BW. It has no-
thing to do with the actual
depth of the orifice below
AB. If an orifice is partly
submerged it must be
divided into two parts, and
only the lower part treated
as submerged. If the
water-level at Y is higher
than at A', as it may be
when XUY is a stream
whose size is not very great relatively to that of the orifice, the
head is BX and not BIF} It is the pressure at A' and not at Y
' Smith's Hydraulics, chap. iii.
Pia. 9.
GENERAL PRINCIPLES AND FORMULAE 15
that affects the discharge from the orifice. The rise from X to F
is owing to the stream being in ' variable flow' (art. 10).
When an orifice is in a horizontal plane, or when it is submerged,
formulae 7 to 10 apply, no matter what the size of the orifice may be.
When an orifice is in a vertical or inclined plane the theoretical
velocity of each horizontal layer of water is \l'2gH, where E is
the head over that layer. When the vertical height between the
upper and lower edges of the orifice is small compared to the
head, the mean velocity in the orifice is practically that at its
centre of gravity. If an orifice extends from M to N (Fig. 9), its
centre being L, it is clear that, the curve KR being nearly straight,
LP is practically the mean of all ordinates from M to N. But
with an orifice HZ, whose centre is F, the protuberance of the
curve UV causes the mean ordinate to fall short of that at F, and
a correction has to be applied depending on the shape of the orifice
and the ratio of its depth to the head over its centre.
6. Flow over Weirs. — Unless the contrary is stated, it will be
assumed that all weirs have vertical side-walls, such forming in
practice the vast majority. The remarks just made regarding the
protuberance of the curve apply a fortiori to a weir. Let M (Fig. 9)
be the level of the crest of a weir. Let AM=H and AS=
9
The mean of all the velocities from A to M is represented by ST}
Thus the theoretical velocity V\%/^ ^6'-q- or' ''- ^2^. The prac-
tical formula is
Q=\d^^E^ . . . (11)
where I is the length of the crest, H the head on the crest, and c
is a co-efficient of discharge whose value for sharp-edged^ weirs
averages abotit '62, and for others varies greatly according to the
form of the weir. With increase of head the co-efficient increases
in some cases and decreases in others. It is not usual to give a
separate formula for finding v or to divide c into c„ and c„ but
roughly these are about the same for sharp-edged weirs as
for sharp-edged orifices. If there is velocity of approach the
formula is
Q='^ cl J2^ {H+nh)i . . . (12)
V
s
where n is I'O or more, and h, as for orifices, is ---, v being the
velocity of approach.
' Fur proof see chap. iii. art. 19.
"' In this paragraph ' sharp-edged ' means ' in a thin wall.'
Fig. 10.
16 HYDEAULICS
When the water on the downstream .side of the weir or ' tail
water' rises above its crest (Fig. 10), the weir is said to be 'sub-
merged ' or ' drowned ' instead
■^ ^ of being 'free.' The discharge
A of JB is found by the ordinary
y^^^ weir formulae, equations 11
_3 and 1 2. The discharge of £0
is considered as being that of
a submerged orifice BC under
vmmm ^ a head AB, and is found by
equation 9 or 10. The tail-
water level should be observed
at L, see remarks concerning submerged orifices (art. 5), but is
often observed at M. The co-efficients used allow for the con-
traction of the stream.
If instead of a weir there are lateral contractions, FGED, the
above equation can still be used, the length I in equation 11 or
12 and the area a in equation 9 or 10 being measured in the con-
tracted part.
In the case, for instance, of a stream in flood the fall AB may
bo small compared to BG in the case of a weir or to BK in the case
of a contracted channel. In such cases equation 9 or 10 alone is
used, and generally 10, since there is usually considerable velocity
of approach. The co-efficients foi such cases are not always
accurately known. See also art. 19.
7. Concerning both Orifices and Weirs. — With all kinds of
apertures small heads are troublesome, not only because of the
difficulty in measuring them exactly, but because complications
occur, and the co-efficients are not properly known.
At a weir the water-surface always begins to fall at a point A
(Fig. 11) situated a short distance upstream of the weir. Hence,
whatever the crest and end contractions may be, there is always
surface contraction. The angular spaces between the wall and
the bed and sides of the channel are occupied by eddies. The fall
in the surface begins where the eddies begin. From this point
the section of the stream proper or forward-moving water
diminishes, its velocity and momentum increase, and the increased
surface-fall is necessary to give the increased momentum (art. 10).
A similar fall occurs upstream of an orifice, though it may only be
perceptible when the orifice is near the surface.
The section where the eddies begin will be termed the ' approach
GENERAL PKINCIPtES AND FORMULA
17
section.' It is here that the head should bo measured and
the velocity of approach observed or calculated, but when, as
often happens
with a weir, and ^^[ ^
generally with an
orifice, the sur-
face upstream of
A is nearly level,
the head may be
observed either
at A or up-
stream of it. It
must not be ob-
served down-
stream of ^. In
some of the older
observations on
weirs the head
was measured from D to C instead
co-efficients thus obtained
but the
is very
of from A to E,
are more variable, and it
difficult in practice to observe the water-level at D with accuracy
The section for velocity of approach may be shifted either way
from AB provided its area is not appreciably altered.
The velocity of approach, v, is the discharge, Q, of the aperture
divided by the area, A, of the approach section. If water enters
a reservoir in such a manner as to cause a defined local current
towards the aperture, the sectional area of the current may be
estimated or observed, and this area, not that of the whole cross-
section of the reservoir, used for determining the velocity of
approach. If the axis of an aperture is oblique to the direction
of the approaching water, the component of the velocity of the
latter parallel to the axis of the aperture may be taken to be the
velocity of approach. Equations 8, 10, and 12 cannot be solved
directly because, until Q or F" is known, v and h are unknown. It
is impossible to find v by direct observation, in the case of a pro-
posed structure or unless the water is actually flowing, and even
then it is not a convenient process. The usual procedure is to
estimate a value for v, calculate h, solve equation 10 or 12, divide
by A, and thus find a corrected value for v. If this differs much
from the value first assumed, it can be substituted and Q calculated
afresh. Velocity of approach has very little effect when the area
of the approach section is about fifteen times that of the smallest
B
18
HYDRAULICS
section of the stream issuing from the aperture, that is for a sharp-
edged aperture nine or ten times the area of the aperture, and for
a bell-mouthed orifice fifteen times the area of the orifice. In a weir
the height of the aperture is to be considered ^E, not DC.
In order that the contraction may be complete the margin must
be clear for a distance from the aperture extending in all directions
to about three times the least dimension of the aperture. Any
further extension has no effect. If the ratio of the width of the
clear margin to the least dimension of the aperture is reduced to
2-67 and 2-0, the discharge is increased by only about '16 and -50
per cent, respectively, so that practically a ratio of 2-75 is sufficient
and will be so regarded. In a weir the length of crest is usually
the greater dimension, and the least dimension is then the head AH.
Another condition which is essential for complete crest con-
traction is that air shall have free access to the space under the
issuing stream. In an aperture in a thin wall with complete con-
traction air usually has free access unless the tail water rises very
nearly to the crest or lower edge, when its surging may shut out
the air. In a weir with no end contractions the width of the
channel, both upstream and downstream of the weir, is, very likely,
the same as the length of the crest, and air will be excluded unle.ss
openings in the side? of the downstream channel are provided to
admit it. Any want of free admission of air causes the sheet of
water to be pressed down by the air above it, the contraction is
reduced and various complications may occur. It is also neces-
sary for complete contraction that the edges be perfectly sharp.
Any rounding increases the discharge.
In Figs. 12 and 13 A BCD is
the boundary of the minimum
clear margin necessary to give
full contraction, supposing iZ/'GiT
to be an orifice, KBCL the boxm-
dary supposing it to be a weir,
and FMXG supposing it to be a
weir with no end contractions. In
Fig. 1 3 EH=EFx 20. The ratios
of the areas within these bound-
aries to those of the apertures are
42-25, 24-38, and 3-75 in Fig. 12,
and 8-29, 4-78, and 3-75 in Fig. 13.
It is thus clear that of the two
conditions, namely, sufl!iciency of the marginal area to give full
GENERAL PRINCIPLES AND FORMULA . 19
contraction and sufficiency of the area of the approach section to
give a negligible velocity of approach, one does not necessarily
imply the other. The two matters must be kept distinct. An
elongated aperture, especially a weir, is most likely to have a high
velocity of approach and a square aperture, especially an orifice, to
have incomplete contraction. Even when the area of the approach
FiQ. IS.
section is very large, it may allow of incomplete contraction in
a portion of an aperture if unsymmetrically situated.
The co-efficients for apertures in thin walls are known with more
exactness than for others, but they are best known for orifices
when the contraction is complete, and for weirs either when it is
complete on all three sides or complete at the crest and absent at
the sides. The co-efficient n for velocity of approach is not very
accurately known. Hence very high velocities of approach are
objectionable where Q has to be accurately computed from assumed
co-efficients, but when v is not very high, that is, when the area
A is more than three times that of the smallest section of the
issuing stream, Q depends very little on n.
The fall in the surface upstream of an aperture, the rise OF due
to crest contraction in a sharp-edged weir, and the eiiect of velocity
of approach greatly complicate the theoretical discussion of weir
formulae.
Section III. — Flow in Channels
8. Definitions. — The 'border,' or 'wet border,' i?, of a stream
is the perimeter of its cross-section, omitting, in the case of an
open stream, the surface width. The 'hydraulic radius,' i2, also
called in the case of an open stream the 'hydraulic mean depth,'
is the sectional area A divided by the border. Thus E—^. The
flow of a stream is ' uniform ' when the mean velocities at succes-
sive cross-sections are equal ; that is, when the areas of the cross-
sections are equal. Otherwise the fiow is 'variable.' A pipe is
20
HYDRAULICS
uniform when all its cross-sections are of equal area. The flow in
such a channel must be uniform when it is flowing full. An open
channel is uniform when it has a constant bed-slope and a uniform
cross-section. The flow in such a channel is uniform when the
water-surface is parallel to the bed, but otherwise it is variable.
The ' inclination ' or ' surface-slope ' of an open stream is the ' fall '
or difference between the water-levels at any two points divided
by the horizontal distance between them. The ' virtual slope ' or
' virtual inclination ' of a pipe is the difference between the levels
of two points in the hydraulic gradient divided by the horizontal
distance between them.
9. Uniform Flow in Channels. — When a stream flows over
a Solid surface the frictional resistance is independent of the
pressure, and approximately proportional to the area of the
surface, and to the square of the velocity. Thus, if / is the
resistance for an area of one square foot at a velocity of one foot
per second, the resistance for an area A and a velocity V is
nearly fA V. The value of / increases with the roughness of
the surface.
In the case of a uniform stream, open or closed, ACDB (Fig. 14),
the second term on the right in equation 5 (p. 11) vanishes, and
the loss of head A in a length
L is equal to the fall in the
surface or in the hydraulic
gradient. In an open stream
the pressures on the ends
AC, BD of the mass of water
are equal, and the accelerat-
ing force is that component of
its weight which acts parallel
to its axis or IFAL-^. On
the assumption that the re-
sistance is entirely due to
friction between the stream
and its channel, tho resistance is approximately /Z^T'-. Since
the motion is uniform this is ecjual to the accelerating force, or
Tr._^f' A h
7 Ti L-
PlO. 14.
But -j^ = B and - =
J3 L
--S, the surface - slope of the stream. Let
GENERAL PRINCIPLES AND FORMULA- 21
or F=CJBS . . . (14)
where C is a co-efficient. In the case of a uniform pipe the pres-
sures on the ends have to be taken into consideration, but the
resulting equation is the same, S being the hydraulic gradient EF.
For if Pi and P^ are the pressures at J. and B, the resultant pres-
sure on the mass ACDB, resolved parallel to its axis, is A{Pi — P^)
or IFA (^-^\ or WA{h'—h). The component of the weight
parallel to the axis is as before WAh. These two together are
WAW. Equation 14 is the usual formula for uniform flow in
streams. It is known as the ' Chczy ' formula. Obviously the co-
efficient is greater the smoother the channel. The formula for
the discharge is
Q=ACsfUS . . . (15).
The theoretical proof just given takes no account of the resist-
ances due to the internal motions of the fluid, nor of the facts that
the velocities at all the different points in the cross-section differ
from one another, that the mean velocity F" of the whole is
greater than the mean velocity v of the portions in contact with
the border, and that the frictional resistance may not be exactly
as V, nor even as «;^ Practically, it is found that the co-efficient
C depends not only on the nature of the channel, but on B and S.
The co-efficient increases with B ; that is, generally with the size
of the stream. It depends also to some extent on S, and perhaps
on other factors which will be mentioned. It increases with S in
pipes of the sizes met with in practice, and in open streams of
small hydraulic radius. The value of C varies generally between
40 and 120 for earthen channels, and between 80 and 160 for clean,
pipes. The chief difficulty with all kinds of channels consists in
forming a correct estimate of the value of C. The difficulty is the
greater because the roughness of a particular channel may be
altered by deposits or other changes.
Let an open stream of rectangular cross-section have a depth
of water D, width W, and velocity F. Let W be great relatively
to D, then B is practically equal to D and the fall in a length
L is ^„ ^ . Let other reaches of the same stream have equal
lengths, but widths 2IF, ZW, etc., the longitudinal slopes being
flatter, so that D is the same in all. The velocities will be
22 HYDRAULICS
— , — , etc., and the losses of head will be , af^T) ' ^**''
The total loss of head in two reaches of widths W and S/F is
-nT-i4y+^)- • The loss of head in two reaches, each of width
2W, will be ZI^(^+|). Thus, the loss of head in a reach of
length 2i and width 2^ is less than half the loss in an equal
length of the same mean width, but in which the width is JV for
half the length and 3/F for the other half. If the streams
compared have circular or semicircular sections the difference is
still greater. Thus, in conveying a given discharge to a given
distance, the advantage as regards fall is on the side of uniformity
in velocity.
10. Variable Flow in Channels. — When the flow is variable,
the loss of head from resistances is the same as in a uniform
stream, that is ^^-j-^, provided the change of section is gradual
and the length L short, so that the velocity and hydraulic radius
change only a little, say by 10 per cent., V And B being their
mean values. Then, from equation 5 (p. 11), the fall in the
surface or hydraulic gradient in the length L is
where Fj and F^ are the velocities at the beginning and end of
the length L. The equation may be written
r=cjTi^'^ • • • (IV)
f^2_ p 2
where h^z= ^S - — --, This is the equation for variable flow in
streams. It is the same as equation 14 (since ;S^=_ ) with the
addition of the quantity A„, which is introduced because of the
change in the vis viva of the water. The quantity F,^ is the square
of the means of all the different velocities in the cross-section. It
ought strictly to bo the mean of the squares. In a case which was
worked out, it was found to be 3-3 per cent, in excess. But a
nearly eijual error occurs with 1'^. The quantity /(„ thus represents
the change of vis viva without appreciable error.
If the section of the stream is decreasing, F, is less than V.,, h is
negative, and V h less than it would be in a uniform stream with
GENERAL PRINCIPLES AND FORMULAE
Fig. 16.
the same values of R and S. Or, V being the same, the fall h in
the surface, or in the hydraulic gradient, is greater than in a
uniform stream. This is because work is being ' stored ' in the
water as its velocity increases. If the section is increasing V^ is
greater than F"., /i, is positive, and V is greater than in a uniform
stream, or V being the same, A is less. Work is being ' restored '
by the water. There may even be a rise in the surface or line of
hydraulic gradient instead of a fall.
Consider any stream AE (Fig. 15) in which the sectional areas
A and E are equal and the velocities therefore equal, and let the
area D be not more
than 10 per cent,
greater than C.
Make C" and G" each
equal to C. Evident-
ly the quantities h„
for the lengths AC, C"E will be equal, but of opposite signs, and
the total fall in the surface in AC -\-C"E will be the same as if
the flow were uniform and the section of the stream were an
average between the sections at A and C. The same is true
of the length O'C and of CG". It does not matter whether the
fluctuations in section are due to changes in the width or in the
depth, or both. The formula V= G J RS therefore applies to a
variable stream AE if the velocities at both ends of it are equal
and the fluctuations moderate, but evidently it does not apply any
the better to a short length of such a stream in which the velocities
at the ends are not equal. Evidently in the stream AE, S varies
from point to point. It is greater as A is less. S in the formula
must be got from the total fall, and G suited to the average section.
Now let the fluctuations be so great that the reaches must be
• subdivided before the equation can be applied to them. Make F
equal to G. The fall in G'F+GG is the same as in a uniform
stream of section S. The fall in FB-\-BG is the same as in a
uniform stream of section K. The total fall in G'G is the same as
the sum of the falls in two uniform streams of sections H and K.
This total fall is (art. 9) grealter than that in a uniform stream,
having a section equal to the mean of H and K. It will also be
seen in section v. that if there are any abrupt changes the falls at
the contractions are by no means counterbalanced by the rises at
the expansions. Thus a variable stream is less efficient than a
uniform stream of the same mean section, or in other words, it
must have a greater total fall in oi'der to carry the same discharge.
24
HYDRAULICS
This and the result arrived at in article 9 are analogous to other
mechanical laws. Uniformity in speed is best, slight fluctuations
are unimportant, but great, and especially abrupt, fluctuations give
reduced efiiciency.
It is clear that the formula V= C JUS applies to the case last
considered if a suitable value is given to C and S is the slope
deduced from the total fall. It even applies approximately to a
stream in which the two end velocities are not equal, provided the
length is considerable, • so that h„ is small relatively to h. It
applies to such a case still more nearly if the value assigned to C
is such as to take account of the change in the end velocity,
0, being greater than for uniform flow if F increases and less if it
decreases. It may not always be easy to say how much C should
be altered in such a case, but it may still be highly convenient
to use the formula in generalising regarding such a stream, for
instance in comparing the discharges for two different water-levels
or stages of supply in an open stream. Thus the formula for
uniform flow applies either exactly or nearly to a vast number of
cases met with in practice in which more or less approximate
uniformity of flow exists.
11. Concerning both Uniform and Variable Flow. — Pipes are
nearly always of approximately uniform section, and the flow in
them nearly uniform, but the sections are seldom exactlj' equal.
Open channels are sometimes nearly uniform and, if there is no
disturbing cause, the flow is nearlj^ uniform. But in both cases
much confusion and error have been caused by applying the
formula for uniform flow to variable streams of short lengths, or,
supposing the short length to be uniform, by carrying the slope
or hydraulic-gradient observations into variable reaches.
Owing to a change, for instance a change of slope, or of section,
or a weir, in a uniform open stream, the water may be 'headed up'
(Fig. 16) or 'drawn down'
(Fig. 17) for a great dis-
tance, AB, upstream of the
point of change. In these
cases (ho surface-slope AB
differs from the bed-slope,
and the flow is variable
although the channel is uniform. Heading-up is also known as
'afllux' or 'back-water.' In all such eases the water-surface ./jS,
which would, if the upstream reach had continued without any
change, have followed the line BO, has to accommodate itself to
Fig. II).
Via. 17.
GENERAL PRINCIPLES AND FORMULA 25
the downstream level at A, and assumes a curve such that the
surface-slope changes in the opposite manner to the sectional
area. Downstream of o .»
A the flow is uniform.
In uniform closed chan-
nels the section of the
stream cannot vary, and
if from any cause the-
gradient-level at any point is altered, the change of slope runs
back to the commencement) of the pipe.
In the absence of any disturbing cause, that is when the flow is
uniform throughout, it is obvious from equations 14 and 15 that
in an open stream an increase of discharge is accompanied by a rise
of water-level and vice versa. The same is the case in a variable
stream. In uniform flow in an open stream, the dimensions and
slope of the channel being known, the discharge can be found if
the water-level is given and vice versa. The surface-slope is the
same as the bed-slope. In variable flow the surface-slope may be
very digerent from the bed-slope, and it is necessary to know the
water-levels at two points in order to find the discharge, or to
know the discharge and the water-level at one point in order to
find the water-level at the other point.
A large stream, whether in an open or closed channel, has an
advantage over a small one both in sectional area and in velocity.
For as A increases B usually increases, and with it C. If the
slopes are equal Q is much greater for the larger stream. If
Q is the same for both, S is much less, that is the loss of head
is less, for the larger stream. This applies to variable as well
as to uniform streams. A fire-hose of diameter D is fitted at its
end with a tapering ' nozzle ' whose least diameter d is perhaps
^-, so that the velocity of the issuing jet is nine times the velocity
o
in the hose. If the hose were made of diameter d the loss of head
in it would be greatly increased, and more pressure would be
required to drive the water through it. The size is limited by
convenience in handling. If part of the hose stretches under
pressure, so that the flow is variable, there is a gain all the same.
Again, let Fig. 16 represent an irrigation distributary with dis-
charge Q, the bed-slope downstream of A being the same as
upstream, so that BC is the water-level. To supply water to high
ground near A a weir may be made, raising the surface to BA,
and enabling a discharge q to be drawn off at A, whereas a small
26 HYDRAULICS
branch made for this purpose from B, with a slope such as BA,
might discharge hardly any water.
The theoretical proof (art. 1) regarding the variation of
pressure with depth depended on the assumption that the velo-
cities at all points in a cross-section were equal. Though they are
not equal, it is found in practice that the law holds good.
12. Relative Velocities in Cross-section. — The velocity at any
point in a straight uniform stream flowing in a channel is,
generally speaking, greater the further the point is removed from
the border. The border retards the motion of the water next to it,
and the retardation is thus communicated to the rest of the stream.
In a pipe of square or circular section the velocity is greatest at
the axis, and thence decreases gradually to the border. In an
open channel the form of cross-section varies greatly in different
streams, and the distribution of the velocities varies with it. The
distribution of velocities in the cross-section of a variable stream,
provided the section of the channel changes gradually, is practi-
cally the same as if the flow were uniform. The distribution
depends on the form of the section, and is not likely to be appreci-
ably affected by the fact that the whole velocity is slowly changing.
In all cases the velocity changes more rapidly near the border
(probably very rapidly quite close to the border, but observations
cannot be made there) and less rapidly towards the centre of
the stream. Thiis all velocity curves are convex downstream.
Nothing in this article relates to the velocities at or near to abrupt
changes of any kind.
13. Bends. — In flow round a bend the distribution of velocities
is modified, the line of greatest velocity being shifted, by reason
of the centrifugal force, towards the outer side of the bend, and
all the velocities on the outer side being increased while those
on the inner side are reduced. The loss of head from resistance
in a bend is greater than in the same length of straight channel.
The additional resistance is chiefly caused by work done in
redistributing the velocities consequent on the transfer of the
maximum line from its normal to its now position, and in the fresh
redistribution after the l)ond is passed. This fresh redistribution
cannot be effected instantaneously, so that the normal distribution
is not restored till some distance below the termination of the
bend. Besides those resistances it is probable that wherever the
distribution is abnormal, no mutter whether any redistribution is
in actual progress or not, the resistance is greater, owing to the
high velocities near the border on the outer side of the bend.
GENERAL PRINCIPLES AND EORMUL^ 27
For a given channel and given radius of bend the total resist-
ance or loss of head caused by the bend is not proportional to its
length because, however long it may be, the redistribution has to
be effected only twice. If the lower half of a bend is reversed in
position, thus forniing two curves, the loss of head in the whole
bend is greater than before, because the redistribution of velocities
has now to be effected in the opposite direction, doubling the
work of this kind done before. No abnormal distribution of
velocities occurs upstream of a bend. The laws regarding bends,
both in pipes and open channels, are imperfectly known. Eecent
experiments on pipes tend to show that, for a given angle sub-
tended by a bend, the actual radius of the bend is, down to a
certain limit, of no great consequence. The only bend which has
any considerable effect is a fairly sharp one. A succession of such
bends may have great effect. Flow round a bend may be either
uniform or variable. If in a sharp bend in an open channel the
section of the stream is the same as in the straight reaches, the
surface gradient must be greater, and there will be heading-up —
though probably slight — in the upstream reach.
Section IV. — Concerning both Apertures
AND Channels
14. Comparisons of different cases. — The difference between the
case of an aperture and that of a channel depends on the nature
of the work done. It is a difference of degree and not of kind.
In flow through a small orifice in the side of a large reservoir
a mass of water which is at rest has a velocity impressed on
it. The motive-power is the pressure of the water due to the
head, and the work done consists almost entirely in imparting
momentum to the water, friction and resistance being unimportant.
In uniform flow in a channel a mass of water slides, under the
influence of gravity, with a constant velocity. The motive-power
is that component of the weight of the water which acts parallel
to the surface or line of gradient, and the work done consists in
overcoming friction and the resistance caused by internal move-
ments. No fresh momentum is iitiparted. These are the two
extreme cases. In flow through some kinds of apertures there
are considerable resistances, and in variable flow in channels much
of the work may consist in the imparting of momentum. The
two extreme cases thus rnerge one into the other.i Most cases of
^ Fig. 10, p. 16, may be regarded as a caae of variable flow.
28 HYDRAULICS
abrupt changes in channels, dealt with in articles 17 to 21, occupy
an intermediate position.
Comparing channels or apertures which entirely surround the
flowing stream with those which leave the water-surface free,
it will be found that the latter are far more elastic than the
former. In the case of the pipe GEF (Fig. 5, p. 9) and the
orifice G (Fig. 9, p. 14), if it is desired to double the discharge, it
is necessary to quadruple the head or the hydraulic gradient. In
either case a very great rise in the water-level AB is required.
But for a weir, since Q is roughly as H'^, in order to double Q
it is only necessary to increase E by some 60 per cent. For an
open channel with vertical sides the discharge — recollecting that C
increases with R — is doubled by increasing the depth about 50
per cent. The above comparisons do not of course take exact
account of variations in the co-efficients. For an open channel
with sloping sides the discharging power may vary very greatly
for a quite moderate change of water-level. When the changes
in the conditions governing the flow are slight, so that the
co-efficient is practically unaltered, the changes in the discharge
are as follows : a change of 1 per cent, in the head over an
orifice or in the slope of a channel changes the discharge '5 per
cent. ; a change of 1 per cent, in the head on a weir or in the
sectional area of a stream changes the discharge 1 5 per cent.
A ' module ' is an arrangement by which it is sought to ensure
a constant discharge of water from a fluctuating source of supply.
Generally it is a machine which automatically alters the size or
position of an aperture as the water-level varies. Some modules
are imperfect, and in such cases, having regard to the preceding
paragraph, it is clearly best that the water to be deli\ere(l should
pass through an orifice or pipe, and the surplus over a weir or
through an open
channel. In Foote's
module (Fig. 18) a
A
s & F I gate E, regulated at
intervals by hand,
causes the water-level
in the canal at G to be
nearly constant, and
higher than at D.
By an orifice F
''"'■ ■'^' • water flows into the
tank FA, and on to the branch AB, the surplus passing over a
GENERAL PRINCIPLES AND FORMULA 29
weir GH. The regulation is better the longer the weir, but it
would be improved by so arranging the gate E that the water
would flow over it instead of under it.
Even if the water in a canal is steady, an outlet consisting of an
orifice of fixed size will not, if submerged, give a constant dis-
charge if the branch channel is liable to be altered. If it is
enlarged, its water-level falls, and thus the head at the outlet is
increased. The limit is not reached until there is a free fall.
15.- Special Conditions affecting Flow. — The condition of water,
as for instance its temperature or the amount of suspended matter
which it contains, has in some cases an effect on the flow. A rise
in the temperature of water probably causes an increase in the
discharge, while an increase in the suspended matter causes, for
flow in channels, a decrease ; but it seems that appreciable changes
in the discharge are caused only by great changes in the con-
ditions, and scarcely even then unless the channels or apertures
are small and the velocities also low.
At very low velocities the nature of flow in pipes is essentially
different from that at ordinary velocities. For any given pipe
there is a certain ' critical velocity.' For velocities lower than this
the motion is in parallel filaments, F varies nearly as S and as R^
and increases with the temperature of the water. When the
velocity rises to the critical amount, a very rapid or even sudden
change occurs, the motion becoming first siniious and then eddying.
The following formulae and figures are approximations. Experi-
ments have been few. For any pipe the critical velocity, F^, is
inversely as .Sthe radius of the pipe. At 0° Cent, it is, for a 1-inch
pipe, about '47 feet per second, for a 12-inch pipe '04 feet per
second. At 100° Cent, the figures are "07 and "OOG, or little more
than -fth of the above. Let F - lower than F,, - be the mean
velocity in a pipe. Then F=361Z)2,S'(1 -f0337y-t- -000221^2),
where D is the diameter of the pipe. If ¥„ is the velocity of
the central filament, V^ = 2 F, and the velocity, V at any radius
R, is Vj, 1 — -jm )■ The kinetic energy of- the water instead of
being slightly in excess of -^ (art. 10) is . If ordinary tur-
bulent motion is artificially produced, stream-line motion re-
establishes itself when the disturbing cause is removed. For any
pipe there is also a 'higher' critical velocity. At 0° Cent, it is,
for a 1-inch pipe, 2 '95 ft. per second, for a ] 2-inch pipe '246 ft. per
second. At 100° Cent, the figures ate -45 and '037. At the higher
30 HYDRAULICS
critical velocity stream-line motion can exist, but a small disturb-
ance upsets it, and once upset it is not likely to re-establish itself.
The subject of critical velocities is not of much practical im-
portance because the velocity in an ordinary pipe or channel is
above Vc, or if it falls as low as Vg the discharge becomes a matter
of little consequence.
16. Remarks. — The solution of a numerical question in
Hydraulics by means of formulae may be either direct or indirect.
When the conditions are given and the discharge, say, is to be
found, it is only necessary to look out the proper co-eflficient and
apply the formula. But frequently the problem is inverted and
consists in finding a suitable set of conditions to give a particular
result. This is especially the case when channels or structures
have to be designed. In many cases a direct solution cannot be
obtained by inverting the formula, either because its form is
unsuitable — an instance of this has been given in article 7 — or
because the co-efficients are not known until the conditions are
determined. It is often necessary to obtain an indirect solution
by assuming a certain set of conditions, calculating the discharge
or other quantity sought, and, if it is not what is desired, making
alterations in the assumed conditions and calculating afresh. In
order to facilitate calculations which would otherwise become very
tedious, numerous working tables are given. By their use work
is vastly reduced.
Both in apertures and channels the co-efficients in the formulae
vary more or less as above stated. Various attempts have been
made to modify the formulae (putting for instance if"', B", S'',
instead of JSi, &, S^) in such a way as to make the co-efficient
constant. Such formulae either have a restricted range or else
the functions of H, R, and S involved are very inconvenient. It
is far better to adhere to the simple indices in common use and
to accept the variations in the co-efficients.
Although for discharge computation one should avoid complex
conditions such as incomplete contraction, small heads, high velo-
city of approach, or variability of flow, yet in practice an engineer
is frequently compelled to accept such conditions, and some atten-
tion will be given to methods of dealing with them.
In many of the more complicated cases (such as some considered
in the following section and in chap, vii.) it may be difficult to
arrive at any exact results by calculation, but it may still be most
useful to recognise the e.x:istence of the phenomena referred to
and to take note of their general effects.
GENERAL PRINCIPLES AND FORMULyE 31
Section V. — Abrupt and other Changes in a Channel
17. Abrupt Changes. — Any change in a channel, whether of
sectional area or direction, and whether or not there is a bifur-
cation or junction, which is so sudden as to cause contraction
or eddies is called an abrupt change. At an abrupt change the
first term on the right in equation 5 (p. 11) is omitted. It
would be small because of the small length of stream considered ;
and owing to the stream being bounded partly by eddies and
changing rapidly in form, it would be difficult to assign values
to the quantities B and G. The second term only is used. Thus
the formuliB are analogous to, or identical with, those for aper-
tures. In fact abrupt changes include submerged weirs and (in
certain respects which will be specially noted) other apertures.
At abrupt changes there are special losses of head, owing to
work being expended on eddies. The length and violence of the
eddies at an enlargement are much greater than at a correspond-
ing contraction (Figs. 3 and 4, p. 5), and the loss of head is
consequently much greater. At a contraction the pressure at
K, L is slightly greater, and in the case of an open stream the
water-level slightly higher than in the flowing stream. These
remarks apply also to orifices and weirs with which there is
velocity of approach. At an expansion the conditions are the'
reverse. The loss of head at an abrupt change of any kind is
most important when the velocity is high ; it can seldom be calcu-
lated with exactness, and often can only be roughly estimated.
18. Abrupt Enlargement. — At an abrupt enlargement (Fig. 4)
the loss of head due to the enlargement can be found theoretically
by assuming that the intensity of pressure on A'G, B'D is the same
as at A'E. Let V^, Ai, be the velocity and sectional area at AB,
Pi the pressure on its centre of gravity, and F^, A^, Pj, similar
quantities at EF. The force A„{Pi—Pi) causes the velocity to be
reduced from V^ to V^. In a short time, t, the fluid ABFE comes
to A'B'F'E. Since the momentum of A'B'FE is unchanged the
change of momentum in the whole mass is the difference between
that of ABE A' and that of EFFE, and that is
\9 9 )
where JV is the weight of a cubic foot of water and Q is the
discharge per second. This change of momentum is equal to the
impulse Ai{Pi—Pi) t, therefore
32 HYDRAULICS
W - g '
But ?^ ~ "' is the fall h in the surface or line of gradient, there-
W
fore from equation" 5 (p. 11)
P —P V'—V
subtracting the precediijg equation from this
(18),
^0 1g _
or the loss of head is the head due to the relative velocity of the
two streams. In order to simplify the calculation it has been
assumed that the stream flows horizontally, that is, that the
centres of gravity of the sections AB, EF are at one level, but
the loss of head due to the enlargement is the same in any case.
The pressure in the eddy has been found to be really less than in
the jet, so that the assumption made is incorrect; and the formula
has been found in practice to give incorrect results for small
pressures and velocities, but for other cases it is fairly accurate.
Equation 1 8 is of the same form as the equation giving the loss
.by shock, in a case of impact of inelastic solid bodies ; and the loss
of head due to an abrupt enlargement is often called ' loss by shock,'
though there is not really any shock, the stream always expanding
gradually.
If there were no loss of head in the length AE there would be
a rise of ^~- — — in the surface or hydraulic gradient. In a
jiipe the loss of head ^- '., — ^^ is always much less than
- ' - " , and there is actually a rise whose amount is approxi-
mately "■ ' -! . . . (1S.\).
This proof is usually given oidy for a pipe, but it clearly apjilies
to an open stream if there is no rise in the surface. If there is a
rise the pressure on the wave QIl, supposing Fig. 4 to be a vertical
section, is not P but P„ (the atmospheric pressure), and the loss
of head is greater than >--L^ -*'-. Moreover, the section usually
changes not only in size but in form, and the redistribution of
GENERAL PRiNCirLBS AND FORMULA 33
the velocities absorbs more work. The rise in the water-level
is thus generally slight, and it cannot usually be calculated
accurately.
When an enlargement is immediately succeeded by a con-
traction so as to cause a deep recess, the water in the recess has
little or no forward motion, and the flow is practically the same as
if the recess did not exist.-
19. Abrupt Contraction. — At an abrupt contraction in a pipe
(Fig. 3) it is necessary, if exact results are required, to calculate
the sectional area at the vena contracta EF and find the velocity
Fj at that section. Then, V^ being the velocity at ST, the fall in
the hydraulic gradient, due to increase in the velocity head from
ST to EF, is — 5__- — 5_, but some head is lost owing to friction
and to the eddies at K, L. The expansion of the stream from
EF to MN causes loss of head, which may be calculated as ex-
plained in the preceding article. The case of an open stream is
analogous, but the whole fall due to loss of head and increase of
velocity head is considered together (art. 6) and equation 10 (p. 14)
is used.
A particular case of abrupt contraction occurs when a stream
issues from a reservoir. There is a fall in the surface or hydraulic
gradient. Most likely the velocity of approach is negligible. If
so the fall, in the case of a pipe, can be calculated without finding
the area EF (chap. v. art. 1), and, if not, the above procedure can
be adopted. For an open stream equation 10 is to be used.
At a local contraction the channel contracts and expands
again, but not necessarily to the same size. For an open channel
equation 10 is used. For a pipe there are various empirical formulae
for local narrowings, all involving the factor — — (chap. v. art. 6).
20. Abrupt Bends, Bifurcations, and Junctions.
— An abrupt bend (Fig. 19) is called an 'elbow. — ~^
The contraction causes a local narrowing of the
stream. It has been found in small pipes that, with
an elbow of 90°, the head lost is very nearly — - pi°' i^-
Judging from analogy and from observation it is probable that this
is nearly true for any pipe and also for an open stream. For elbows
of other angles the relative loss of head is knov.'n for small pipes
(chap V. art. 6), and it may be assumed that for other channels it
is roughly the same.
CI
34
HYDRAULICS
Fio. 21.
At a bifurcation (Figs. 20 and 21) the stream entering the
branch may be regarded as flowing round a bend whose outer
boundary is shown by
~^ ^ the dotted lines. In the ~^ -.
main channel below the
branch there is an en-
largement (art. 18). Let
be the angle made by the centre lines of
the branch and of the main channel upstream of it. When
6 is 90° or thereabouts the whole head due to the velocity is
lost, and there is a fall in the surface or hydraulic gradient
of the branch of about the same amount as there would be
if it issued from a reservoir. But if V is high the absence of
contraction at A does not compensate for the excessive contraction
at B, and the fall is increased, or the discharge of the branch
diminished. When exceeds 90° the component of V resolved
parallel to the axis of the branch may be regarded as velocity of
approach, the discharge being increased accordingly. It is not
known for what angle the velocity of approach compensates for
the greater contraction as compared with that in the case of a
reservoir. The angle differs with the velocity and probably with
the width of the branch, and is perhaps generally not much
greater than 90°. By
the arrangement shown
in Figs. 22 and 23, the
losses of head both in the
branch and in the main
stream are reduced, and
that in the branch is not relatively altered by a high velocity.
, the branch is 'bell-
— ^ mouthed' (Figs. 24
~ and 25) the loss of
head in it is some-
what reduced, and
it is further re-
duced by filling in
the portions shown in dotted lines, thus doing away with eddies.
Figs. 20 to 25 represent junctions if the stream is supposed to
flow in the directions opposite to those of the arrows. The losses
of heud are very much the same as in the corresponding cases of
bifurcations.
21. Concerning all Abrupt Changes.— The ' limits ' of an abrupt
Fio. 22.
If
Fin. 24.
Fio. 25.
GENERAL PEINCIPLES AND FORMULA 35
change are those of the peculiar local flow caused by it. The
upstream limit is, in Fig. 4, at A'B', in Fig. 3, just as with a weir
and certain kinds of orifices (art. 7), at ST. In the other cases it
is where the eddying or curvature begins. In all cases eddies
exist in the stream itself for some distance downstream of an
abrupt change. The downstream limit is where these eddies have
become reduced. They may not cease altogether for a long
distance.
In the reach downstream of an abrupt change the flow, except
for eddying and probably disturbance of the relation to one
another of the various velocities in the cross-section, is normal,
and the water-surface or hydraulic gradient takes the level suited
to the discharge just as if no abrupt change existed. Within the
limits of the abrupt change there occurs the fall or rise discussed
in the three preceding articles. Thus the level of the surface or
hydraulic gradient at the downstream limit of the abrupt change
governs that at the upstream limit, and this again affects the slope
in the upstream reach in the manner indicated above (art. 11).
But the distribution of the velocities in the upstream reach is
normal.^ There is nothing to affect it until the abrupt change
actually begins. (Cf. also Bends, art. 13.) Thus, at all changes,
whether of sectional area or direction of flow, and whether strictly
abrupt or not, the effect on the hydraulic gradient or slope
is wholly upstream, but eddies and disturbance of the velocity
relations are wholly downstream.
It follows that discharge observations in which the mean
velocity of the whole stream is to be deduced from observations
taken, say, in the centre only, should not be made within a con-
siderable distance downstream of an abrupt change, but may be
made a short distance upstream of it.
Any alteration which makes a change less abrupt reduces the
loss of head. This has been seen in considering bends, elbows,
and bifurcations. Regarding changes of section an instance would
be the rounding of the edges of the weir in Fig. 10, p. 16, or the
addition of long slopes upstream and downstream. It has been seen
(art. 10) that in a short channel which gradually alters in section
and then reverts to its former section, the gain of head is equal to
the loss. In an open channel there will be a slight local hollow in
the surface or a protuberance on it. The hollow can often be seen
over a submerged weir which has gradual slopes. In any case the
loss of head is negligible if the change is gradual, and especially if
it is free from angularities.
' Haying regard to the altered cross-section. See art. 12.
36 HYDRAULICS
Section VI. — Movement of Solids by a Stream
22. Definitions. — When flowing water transports solid substances
by carrying them in suspension, they are known as ' silt.' Water
also moves solids by rolling them along the channel. The weight
of silt present in each cubic foot of water is called the ' charge ' of
silt. Silt is chiefly mud and fine sand; rolled material is sand,
gravel, shingle,, and boulders. When a stream obtains material by
eroding its channel, it iS said to ' scour.' When it deposits material
in its channel, it is said to ' silt.' Both terms are used irrespective
of whether the material is carried or rolled. Material of one kind
may be rolled and carried alternately.
23. General Laws. — It is well known that the scouring and trans-
porting power of a stream increases with its velocity. Observations
made by Kennedy prove that its power to carry silt decreases as
the depth of water increases. ^ The power is probably derived from
the eddies which are produced at the bed. Every suspended
particle tends to sink, if its specific gravity is greater than unity.
It is prevented from sinking by the upward components of the
eddies. If V is the velocity of the stream and D its depth, the
force exerted by the eddies generated on one square foot of the bed
is greater as the velocity is greater, and is, say, as F". But, given
the average charge of silt, the weight of silt in a vertical column
of water whose base is one square foot is as J). Therefore the
power of a stream to support silt is as F" and inversely as D.
Kennedy found that for the heavy mud mixed with fine sand found
in the rivers of Northern India — except in their low stages — where
they debouch from the Himalayas,
F= •84i)-64 ... (19)
This equation is not exact. It is impossible to construct a theo-
retical equation which shall include both suspended and rolling
matter, because the proportions in which they exist are not
known.
A stream of given velocity and depth can only carry a certain
charge of silt. When it is carrying this it is said to be 'fully
charged.' In this case, if there is any reduction in velocity, or if
any additional silt is by any means brought into the stream, a
deposit will occur (unless there is also a reduction of depth) until
' Min. Proc. Inst. C.E., vol. oxix.
GENERAL PRINCIPLES AND FORMULA 37
the charge of silt is reduced again to the full charge for the stream.
The deposit may, however, occur slowly, and extend over a consider-
able length of channel.
The full charge is affected by the nature of the silt. The
specific gravity of mud is not much greater than that of water,
while that of sand is about 1-5 times as great. The particles of
sand are larger. If two streams of equal depths and velocities
are fully charged, one with particles of mud and the other with
particles of sand, the latter will sink more rapidly and will have to
be more frequently thrown up. They will form a smaller propor-
tion of the volume of water. '
It is sometimes supposed that the inclination of the bed of a
stream, when high, facilitates scour, the material rolling more
easily down a steep inclined plane. The inclination is nearly
always too small to have any appreciable direct effect on the rolling
force. In fact the' bed is generally more or less undulating, and
the movement may be either uphill or downhill. The inclination of
the surface of the stream of course affects its velocity, and this
is the real factor in the case.
It has sometimes been said that increased depth gives increased
scouring power, because of the increased pressure, but this is not
so. The increased pressure due to depth acts on both the up-
stream and downstream sides of a body. It is moved only by the
pressure due to the velocity.
To what degree the addition of a charge of silt to a pure stream
affects its velocity is not known. It is not likely that it has any
appreciable effect.
If a stream has power to scour any particular material from its
bed, it has power to transport it ; but the converse is not usually
true. If the material is hard and compact tlie stream may have
far more difficulty in eroding it than in transporting it.
If a stream is not fully charged, it tends to become so by scour-
ing its bed. A stream fully charged with mud cannot scour mud
from its bed, but its power to roll solids is, perhaps, unaffected by
its being charged with mud.
In the ' Inundation Canals,' so called because they flow only
when the rivers are in flood, fed from the rivers of Northern India,
the silt entering a canal usually consists of sand and mud. The
sandy portion, or most of it, is deposited in the head reach of the
canal, forming a wedge-shaped mass, with a depth of perhaps two
or three feet at the head of the canal, diminishing to zero at a
point a few miles from the head. Beyond this point the water.
38 HYDEAULICS
charged with mud and perhaps a little sand, usually flows for many
miles without any deposit occurring, although there are frequent
reductions in the velocity caused by the diminutions in the size of
the stream as the distributaries are taken off, and sometimes also
by reductions in the gradient. The absence of further deposits,
inexplicable till the discovery of Kennedy's law, is due to the
fact that the depth of water diminishes as well as the velocity.
Many of the channels were constructed long ago by the natives,
and they seem to have learned from experience to give the
channels such widths that the depth of water decreases at the
proper rate.
It is a common practice to so reduce the velocity of a stream
that silting must take place. The object may be either to
' warp up ' certain localities by silt deposit or to free the water
from silt, and thus reduce the deposit in places further down.
When the velocity of a stream is arrested altogether, as it practi-
cally is when a stream flows through a large reservoir, the whole
of the silt will deposit if it has time to do so, that is, if the reser-
voir is large enough. Low-lying and marshy plots of ground may
be silted up, and rendered healthy and culturable by turning a
silt-bearing stream through them. In order to prevent deposit in
the head of a canal the water may be made to pass through a
' silt-trap ' or large natural or artificial basin, where the velocity is
small, or the supply may be drawn from the upper layers of the
river water (art. 24).
Silting and scouring are generally regular or irregular in their
action according as the flow is regular or irregular, that is, accord-
ing as the channel is free or not from abrupt changes and eddies.
In a uniform canal fed from a river the deposit in the head of the
canal forms a wedge-shaped mass, as above stated, the depth of
the deposit decreasing with a fair approach to uniformity. Salient
angles are most liable to scour, and deep hollows or recesses
to silt. Eddies have .1 strong scouring power. Immediately
downstream of an abrupt change scour is often severe.
Most streams vary greatly at different times both in volume
and velocity and in the quantity of material brought into them.
Hence the action is not constant. A stream may silt at one
season and scour at another, maintaining a steady average. When
this happens, or when the stream never silts or scours appreciably
it is said to be in ' permanent regime'
Waves, wliethcr due to wind or other agency, may cause scour,
especially of the banks. Their efiect on the bed becomes less as
GENERAL PRINCIPLES AND I'ORMULJi; 39
the depth of water increases, but does not cease altogether at a
depth of 21 feet, as has been supposed. Salt water possesses
a power of precipitating silt.
24. Di-stribution of Silt Charge.— Since the eddies are strongest
near the bed, the charge of silt must generally increase towards
the bed, but the rate of increase varies greatly. Mud having
a low specific gravity, the charge is probably nearly as great near
the surface as elsewhere. Sand ' is heavy, and is oftener rolled
than carried. "When carried it is usually in much greater propor-
tion near the bed. Materials, such as boulders, do not generally
rise much above the bed. A perfectly clear stream may be rolling
solids. The ratio of the silt-charge at the surface to that at the
bed thus varies from to 1. For a given kind of silt the rate
of variation from surface to bed probably increases with the
depth and decreases with the velocity. The distribution in any
particular stream can only be ascertained by observation, or by
experience of similar streams. It is a matter of great practical
importance, as affecting the best bed-level for a branch taking off
from the stream. The results of observations show considerable
discrepancies, even when averaged, and individual observations
very great discrepancies. In some rivers 10 to 17 feet deep the
silt charge has been found to increase at the rate of about 10
per cent, for each foot in depth below the surface. In others,
with depths ranging up to 16 feet, the silt charge at about three-
fourths or four-fifths of the full depth has been found to bear to
that near the surface, a ratio varying from 1;^ to 2.
Section VII. — Hydraulic Observations
AND Co-efficients
25. Hydraulic Observations. — It is frequently necessary in
Hydraulic Engineering to observe water-levels, dimensions of
streams, and velocities, and from these to compute discharges.
The object of a set of observations may be either simply to
ascertain, say, the discharge in a particular instance, or to find
and record the co-efficients applicable to the case, so as to enable
other discharges under similar conditions to be calculated. Obser-
vations of the latter class, when extensive, are usually termed
'Hydraulic Experiments.' A consideration of the instruments
and methods adopted in Hydraulic Observations may be strictly
a matter of Hydraulic Engineering, but it is necessary to include
it in a general manner in a Trea.tise on Hydraulics, both because
40 HYDRAULICS
the principles involved in such work are closely connected with
the laws of flow, and also in order that proper estimates may be
formed of the errors which are possible and of the reliability of
the results which have been arrived at by various observers.^
In making observations accurate measurements of lineal dimen-
sions, depth, and water-levels are necessary, as well as accurate
timing. The number and duration of the observations should
be sufficient to eliminate the efFects of the irregular motion of
the water, and bring out the true average values of the quantities
sought for. Owing to imperfections in these matters, or in the
instruments used, errors of various kinds may occur. These are
known as 'observation errors.' They may balance one another
more or less, but are liable to accumulate in one direction in a
remarkable manner. Care in observing, as well as sufficiency in
the number of observations, are therefore essential points. An
error in measuring length or time has, of course, a greater relative
effect when the amount measured is smali. In a channel the fall
in the surface or hydraulic gradient is often a small quantity, and
thus in slope observations the error is often large. With an aper-
ture under a small head the error in observing it may be serious.
It has been shown by Smith ^ that, even in the careful experiments
made by Lesbros on orifices, the co-efficients were probably affected
by such causes as the expansion and contraction of the long iron
handles attached to the movable ' gates,' and to the bending, under
great pressure, of the plates forming the orifices. Besides quantities
which can be actually measured there are conditions which can
be observed but may be overlooked, such as a slight rounding
of a sharp edge, the clinging of some portion of the water to an
aperture when it is supposed to be springing clear, or the occurrence
of a deposit in a channel. Such matters not always very perceptible
may have considerable effects on the flow.
Again, there are conditions which cannot be ascertained, and
assumptions are made regarding them. It has, for instance, been
assumed that a local surface-slope too small to bo observed is the
same as the observed slope in a great length, or that the diameter
of a pipe, measured at only a few places, is constant throughout.
Lastly, there are some things very difficult to descril)e, such as
the degree of sharpness of an edge, or of roughness of a channel.
Thus there is often, in accounts of experiments, a defective or
erroneous description of the conditions which existed. This
may Ix) termed 'descriptive error.' In some cases it has been
1 Details will bo given in chap. viii. ^ Hydraulics, chap. iii.
GENERAL PRINCIPLES AND FORMULA 41
very great. Its effect is similar to that of observation error, and
the line between the two cannot easily be drawn.
When the quantity whose law of variation is sought depends
on several conditions which vary together, it is often difficult to
determine the effect of the variation of any one condition alone.
As far as possible observations should be made with only one con-
dition varying at a time. Generally, observations at one site are
kept distinct from those at other sites, but if the conditions of
different sites are nearly similar, it is legitimate to combine observa-
tions at different sites. In such a case, care should be taken that
the effect of any slight or accidental dissimilarity in the sites will
not affect any one set of values, but will be distributed throughout
all. It wouldj for instance, be undesirable to have all the low-
water observations at one site and the high-water observations
at another.
A series of observations containing a source of error may show
results quite consistent with one another, and may be of great use
in bringing out certain laws. The well-known weir experiments
of Francis and of Fteley and Stearns give results which are con-
sistent, and were for long accepted as practically correct ; but
when they are compared with the later results of Bazin certain
discrepancies appear, and it is clear that one or the other set of
experiments contains some error.
Detailed accounts of Hydraulic Experiments do not, of course,
find a place in a textbook. References to the chief works on such
experiments have already been given (p. 7), but special points will
be noticed whenever necessary.
26. Co-efficients. — From the causes above stated the co-efficients,
or other figures, arrived at by various observers frequently show
grave discrepancies. This is especially the case with- the older
experiments. In the more recent ones the discrepancies have been
reduced.
The ' probable errors ' of co-efficients have in some cases been
estimated by those who have investigated them. The meaning of
this may be explained by an example which will be made to
include all kinds of errors. Let a weir have a crest 1 foot wide,
sharp edges, and a head of 1 foot. Suppose the co-efficient arrived
a,t is -600,, and that it is estimated that the observation error may
probably be 1 per cent, either way. Then 1 per cent, is the
probable error, and the value of the co-efficient is as likely to
be between -606 and -594 as to be outside of these limits. But
there may also have been descriptive errors connected with, say,
42 HYDRAULICS
the width of the crest or sharpness of the edges, and the reax
probable error may be much greater than 1 per cent. Finally, if
the co-efficient is applied to a weir, over which water is actually
flowing, there may be again observation error in measuring the
head. Sometimes these different errors balance one another,
but sometimes, as before remarked, they all accumulate in one
direction.
The co-efficients for different cases contain probable errors of
very different amounts. For thin-wall apertures under favour-
able circumstances, the probable error is only about "50 per cent.
For channels and especially for pipes, owing chiefly to the causes
above indicated (arts. 9 and 11), it may easily be 5 or 10 per cent.
Although in the above instance the final operation of observation
introduces an additional errot, complete observation is much
better than calculation. If no co-efficient had been assumed at
all, but the discharge of the stream carefully observed, as well as
the head on the weir, then both the discharge and the co-efficient
for that particular case would have been obtained in the best
possible manner.
The results of individual experiments nearly always show irregu-
larities, that is when plotted they do not give regular curves.
The usual method is to draw a regular curve in such a manner as
to average the discrepancies and correct the original observations.
Most published co-efficients have been obtained in this manner.
When an experimenter obtains a series of co-efficients for any
particular case, he often connects them by an empirical formula
involving one or two constants. This has been done by Bazin
and Kutter for open channels, and by Fteley and Stearns, Francis
and Bazin for certain kinds of weirs, ^^^lat the engineer really
needs and uses is a table of the co-efficients, but the formulae may
be useful in finding a co-efficient when a table is not at hand, or in
finding its value for cases intermediate between those given in the
tables or outside the range of the observations. This last practice
must, however, be adopted with caution and within narrow limits.
Further experinients are required in all branches of hydraulics.
A feature in future experiments will no doubt be the increased
use of automatic and self-recording methods.
The most recent observations generally command most confidence.
Causes of error are constantly being studied and eliminated. Due
weight is given to this consideration in the task — often difficult —
of deciding what figuresshall be adjudged to be the best.
CHAPTEE III
OEIFICES
[For preliminary information see chapter ii. articles 4, 5,- 7, 14, and 15]
Section I. — Orifices in General
1. General Information. — The jDrincipal kinds of orifices or
short tubes met with in pra,ctice, with their average co-efficients,
are as follows : —
Sketch.
Fig. 26.
Pig. 32.
Pig. 25a.
Fig. 27.
Orifice in thin wall,
Bell-mouthed tube,
Convergent conical
tube.
Cylindrical tube, .
Inwardljr project-
ing cylindrical
tube,
Borda's mouth-
piece.
Divergent conical
tube.
Divergent tube
with bell-mouth,
* For the smaller end of the tube and when angle of cone is 13'
t For the smaller end when angle is 5" 6'.
t For the smallest section.
Fig. 29.
Fig. 31.
Description of Orilice
or Tube.
Average Co-efFicients
for Complete
Contraction.
Cc
C„
c
•63
■97
•61
1-0
•97
•97
•98*
•96*
■94*
VO
•82
•82
ro
■72
•72
•52
■98
■51
r46t
ro|
S'OJ
2-Ot
is 13°.
44 HYDRAULICS
The co-efficients given, except for conical tubes, are approximate
and average values, further details being given in the succeeding
articles. The length of a tube must not exceed three times the
diameter; otherwise the co-efficient is reduced, owing to friction,
and the tube becomes a pipe. A tube generally has its axis hori-
zontal, but may have it in any direction. If the lengths of the
cylindrical tubes (Figs. 28 and 29) are reduced till the jet springs
clear from the upstream edge, the co-efficients change to the values
shown for Figs. 25a and 30. The length at which the change takes
place may for a very great head be two diameters or more, but is
generally less than one dia.meter. The cross-sections of all the
tubes are supposed to be circular, but the co-efficients apply nearly
to square sections and to others differing not greatly from circles
and squares. Thus ' cylindrical ' includes ' prismatic,' and similarly
with the others. In the case of an elongated section, ' diameter '
is to be understood as 'least diameter.'
For orifices up to a foot in diameter, metal edges filed sharp
should be used, if full contraction is required. For larger orifices
edges of wood, stone, or brick give fair accuracy. These remarks
apply to all kinds of orifices in which the edges are supposed to
be sharp, that is to all except bell-mouths, though with a con-
vergent conical tube the effect of want of sharpness is probably
small, the final contraction occurring outside the tube.
In the cases of the inwardly projecting tubes represented by
Figs. 29 and 30, the tubes are supposed to be quite thin and their
inner edges sharp.
The co-efficient of discharge does not generally alter much as
the head varies, so that, neglecting the efiect of velocity of
approach, the discharge through a given orifice under different
heads is nearly as R^. In order to double the discharge H must
be quadrupled. If the head is doubled the discharge is increased
in the ratio of about 1-4 to 1.
To facilitate the working out of problems, the theoretical veloci-
ties corresponding to various heads are given in table i. V can
bo found from H or H from V.
2. Measurement of Head. — Upstream of an orifice there may
be a vortex in the water, or, when the velocity of approach is high,
ORIFICES
45
ta
$in.
Fig.
a wave or heaping of water where it strikes the wall, and the
head should be measured a short distance upstream from such
vortex or wave. If the part of a reservoir adjoining an oriiice is
closed (Fig. 33) the head may be
measured at R, but if the length of
the closed portion is more than thrice
its least diameter, it is necessary to
find the loss of head in it, treating it
as a pipe.
Smith states that for an orifice in a
thin wall the head should probably be measured to the centre of
gravity of the vena contracta. The matter seems to admit of no
doubt, and the rule should apply to all kinds of orifices in which
there is contraction. It is at the vena contracta and not elsewhere
that the theoretical velocity is JlgU. In a bell-mouthed orifice
in a horizontal wall the head would be
5=; measured to the ' discharging side ' of
the orifice, and the jet from an orifice
in a thin horizontal wall issues under
the same conditions, except that friction
against the sides is removed. Under
a small head the jet from an orifice in
a thin vertical wall may drop appreci-
ably in the distance PM (Fig. 34), and
the true head, that at M, is not the
same as at P, the centre of the orifice.
Nearly all co-efficients have been ob-
tained from orifices in vertical walls
under considerable heads, so that it
has made no difference how the head has been measured ; but in
applying these co-efiicients to orifices in other
positions the head should be measured to the
vena contracta.
3. Incomplete Contraction. — The contrac-
tion in an orifice with a sharp edge may be
partly suppressed by adding an internal pro-
jection AB (Fig. 35), extending over a portion
of the perimeter of the orifice. The con-
traction is then said to be 'partial.' If the
length AB is not less than 1 '5 times the least
diameter of the orifice, the co-efficients for orifices in thin walls
are, according to Bidone —
Fio. 34.
Fio. 35.
46 HYDRAULICS
For a rectangular orifice r^,=cn4- -152- j . . . (20),
For a circular orifice c^=cfl + •128-^1 . . . (21),
where c is the co-efiicient of discharge for the simple orifice, P its
perimeter, and S that of the portion on which the contraction is
suppressed. Partial suppression may be caused by making one or
more of the sides of an orifice flush with those of the reservoir.
The above formulae were obtained with small orifices and heads
S
under six feet. They are not applicable when p- is greater than
f for a rectangle or | for a circle. They are not quite reliable in
any case, and especially when the orifice is elongated. With a
rectangular orifice of length twenty times its breadth the suppres-
sion of the contraction on one of the long sides has been found
to increase c by 8 to 12 per cent., whereas by the formula the
increase should be 7 '2 per cent.
The table on p. 56 shows that suppression of the contraction
on 1, 2, 3, and 4 sides of an orifice 4 ft. square caused c to increase
by about 4, 13, 28, and 56 per cent, respectively, the final result
(c about '25) being very much what would be expected.
If the contraction is suppressed on part of the perimeter, that
on the remaining part increases, and this is what would be ex-
pected. The increase is, no doubt, most pronounced on the side
opposite to the suppressed part, because the contracting filaments
of water are no longer directly opposed by others.
In a bell-mouthed tube the contraction must be complete, what>
ever the clear margin may be. In all other cases decrease in the
clear margin causes the contraction to be 'imperfect.' In chapter
iv. (art. 3) some rules are given regarding the allowance to be
made for imperfect contraction with weirs in thin walls. Con-
sidering them in connection with the above formula} for partial
contraction the figures shown in table ii. are arrived at. In this
table S' is the length of the perimeter on which the clear margin
is reduced, G the width of the margin in the reduced part, d the
least diameter of the orifice, and c, Ci the co-efiicients for the orifice
with complete and incomplete contraction respectively. The
"table is meant for orifices in thin walls, but even for these it is
only approximate. The table on page 56 deals with some other
orifices with sharp edges. The above formulie and figures apply
to Co as well as to c, both probably altering in about the same pro-
ORIFICES
47
portion and c, being constant. It may happen that the contraction
is suppressed on one part of the perimeter of an orifice and
imperfect on another part. Example 4, page 74, shows the
method which may be adopted for such cases. When the con-
traction is either suppressed or very imperfect on nearly the whole
perimeter the approximation becomes very doubtful.
When an orifice '30 feet long and '05 feet high was bisected by vertical
brass sheets of various thicknesses, it was found that a very thin sheet had
little or no efifect either on c or on the jet, but a sheet "04 feet thick increased
c nearly 1 per cent., the jets, however, uniting a short distance from the
orifice.^
4. Changes in Temperature and Condition of Water. — The
results of some experiments by Smith, Mair, and Unwin re-
spectively are shown in the following table : — ^
Kind of
Orifice.
Dia-
meter.
Amount
by wliicli
Tempera-
ture of
Water was
raised.
Effect on the
Discharge.
Head.
Remarks.
Orifice in thin
wall.
Bell-mouthed
tube.
Inches.
■ -24
. -40
2-5
' -40
" 1-5
Fahr.
82°
144°
96°
110°
115°
Decrease of 1 1
per cent.
Decrease of 1
per cent.
Increase of ^
per cent.
Increase of 3J
per cent.
Increase of 2
per cent.
Feet.
•56 to 3-2
1 to 1-5
1-75
1 to 1-5
1-75
In all cases
the initial
temperature
of the water
was normal,
namely, 45°
to erFahr.
It is clear that it requires a great change of temperature to cause
an appreciable change in the discharge, and that the change is
greater the smaller the orifice. The law governing the change is
not clear. Smith considers that with a head of 10 feet a change
of 50° in temperature probably has no appreciable effect for orifices
of more than -24 inch in diameter.^
Smith states that for small orifices ('05 foot and less in diameter,
and with heads less than 1 foot) the discharge fluctuates consider-
ably, and that this is perhaps due to unknown changes in the
character of the water. With either larger heads or larger orifices
' Smith's Hydraulics, chap. iii.
^ Ibid, and Min. Proc. Inst. C.E., vol. Ixxxiv.
^ Hydraulics, chap. iii.
48 HYDRAULICS
the uncertainty disappeared. It was not due to experimental
error.
Smith also states as follows. Water containing clayey sediment may
have a greater oo-efEoient because of its oiliness. Thick pil, though very
viscous, has a greater co-efficient than water. When the water is in a
disturbed condition, and approaches the orifice in an irregular manner, the
jet may be ragged and twisted, but c is not afifected appreciably. Greasy
matter adhering to the edge of an orifice slightly reduces the discharge, if
the diameter is '10 foot or less, the reduction being due to the diminished
size of the orifice.
5. Velocity of Approach. — The subject of velocity of approach
is of more importance for weirs than for orifices, and a full discus-
sion regarding it is given in chapter iv. (art. 5). In equations 8
and 10 (pp. 13 and li) n may be taken to be I'O, when the
aperture is opposite that part of the approach section where the
velocity is greatest — that is generally the central part and near
the surface— and about -80 when it is opposite a part where the
velocity is lowest — that is near the side or bottom.^ The method
of solving the above equations has been stated in chapter ii.
(art. 7). For an orifice with sharp edges, whenever velocity of
approach has to be taken into account, there will very likely be
imperfect contraction on some part of the perimeter, and f, must
be substituted for c.
Another method of procedure is to alter the forms of the equa-
v' a'' V-
tions. Since A.= -~=-j2 . -— therefore equation 8 may be
written V^=c^'('lgH+n^\r'\.
Whence vU- c^- • « . ^^ = c/ . 2gE.
Or V=c,j2^ /- 7-^ .
(22).
And Q=<'-a-^/^gH /' --ff' ■ . . (23);
1
These can bo solved directly. The quantity / . „ a"- is 'a
V ^ '-ll""— To
A'
co-eflicient of correction ' for velocity of approach. It may be
denoted by c„. Table iii. shows some values of -~, for difierent
' But there are then (art. 3, also pp. 18, 19) disturbing factors. Practically
n is taken as 1 '0.
2
For other forms of this equation see chap. viii. art. 17.
ORIFICES 49
values of - , and it also shows the value of c^ and of the quantities
leading up to it, for c^=-97 and n=l-0. For a bell-mouthed tube
a' is simply the area of the discharging side of the tube and c„ is c.
When -^ is less than -- a change in c^ or in n makes very little
difference in c„, and a mere inspection of the table will enable its
proper value to be found. Thus the use of c^ simplifies matters.
For other kinds of orifices c must be separated into its factors
Cj and Cy, and a' found by multiplying a by c^. But it will be seen
from the- examples (p. 72 et seq.) that the use of c„ may
often be convenient. In all cases the use of c„ causes a little
A
inaccuracy when — is small. If greater accuracy is required c„
may be used for the first approximation only. Another form of c„ is
/ 1 _ 2 a' , which would be very convenient for sharp-edged
orifices, but there are so many values of c that extensive tables
would be needed.
Let cCa=C, then is an 'inclusive co-efficient' and
Q=Caj2gri . . . (24).
This formula is not convenient for general use, because it would
be difficult to tabulate all the values- of for different kinds of
orifices for various velocities of approach. Bu.t where it is
desired to ascertain by experiment the co-efficients for any orifice,
so as to frame a, discharge table for that orifice alone, then equa-
tion 24 is by far the best and simplest to use.
If there are two orifices supplied from the same reservoir and
situated not far apart, the discharge of each may be increased by
the effect of the other, especially when both are in the same wall.
In Bazin's experiments twelve orifices, each 8" x 8" nearly, and
capable of being closed by gates, were placed side by side. The
following values of the inclusive co-efficient C were found : —
Number of gates open : 1 2 3 4 5 or more.
Total co-efiicient for all : "633 "642 -646 '649 -650.
When one gate was raised two inches and the others were fully
opened the co-efficients were as follows : —
Number fully open : 1 2 3 4 5 or more.
Co-efficient for the one \ .g^Q .g^^ .gg^ .gg^ .ggg_
partly open : )
The contraction was not complete, the twelve orifices being in
D
50 HYDRAULICS
the end of a chamber only 18 feet wide. In order that two
orifices in the same plane may have no effect on one another, it
is probable that there should be no overlapping either of the
minimum clear margins or of the minimun? areas of approach
sections requisite for full contraction and for negligible velocity
of approach respectively (cf. chap. v. art. 2).
6. Effective Head.— The 'effective head' over an orifice is the
head which would produce the actual velpcity supposing c^ to be
unity. If H and H^ are the actual and effective heads
V=c,j2iH= sl^E, . . . (25).
If H—H,=:Hr, then Hr is the head wasted in overcoming resist-
TT
ances. Let rf-='^r, then Cr is the 'co-efficient of resistance,' or
ratio of the wasted to the effective head.
Since l+c,=^^+^'-=f.
77" 1
And from equation 25 -==-= — j.
Therefore «,.=—,_ _l . . . (26).
If there is velocity of approach H-\-nh must be put for H in
the foregoing. The following table shows the values of c^ for
different values of c„. The head wasted is only a small per-
centage of the effective head, when c^ is high, but it may be more
than the effective head when r„ is low.
c„=-995
■99
•98
■^■7
•95
•90
Cr=-010
•020
■041
•063
•111
•233
c„=-85
•82
•80
•75
•72
•715
■70
c, = -384
•489
•563
•778
•929
•956
1-049
The equation V= JlgH^ gives the actual velocity for an orifice
referred to an imaginary water-surface situated Zf,. feet below the
actual surface (Fig. 40), but the equation will not apply to another
similar orifice in the same reservoir at a different level, because
Hr will not have the same value.
7. Jet from an Orifice. — The jet of water from an orifice retains
its coherence for some distance and then becomes scattered.
With an orifice in a thin wall, not circular and not in a horizontal
plane, and with a head not very great compared to the size of
the orifice, a phenomenon called 'inversion of the jet' occurs.
The section of the jet is at first nearly of the shape of the orifice,
ORIFICES
51
Fig. 36.
but afterwards spreads into sheets perpendicular to the sides of
the orifice. Those portions of the jet which issue under different
heads behave somewhat similarly to separate jets, which, if two
of them meet obliquely, spread into a sheet perpendicular to the
plane containing them. This expansion into sheets reaches a
limit, and the jet contracts again to nearly the form of the orifice,
but if its coherence is retained it again
throws out sheets in directions bisect-
ing the angles between the previous
sheets. This is probably due to sur-
face tension or capillarity. The fluid
is enclosed in an envelope of constant
tension, and the recurrent form of the
jet is due to vibrations of the fluid column.^
Fig. 36 shows the cross-sections of jets from two square orifices,
the orifices being supposed to be far apart.
r->. At a corner the two streams A and G
in contracting interfere with one another,
and some fluid is forced towards the
corner. The full line in Fig. 37 shows the
form next assumed, and the dotted line
that assumed subsequently. The dotted
lines in Fig. 36 show the form of jet
where the two squares are joined to
form a rectangular orifice.
Let H^ be the effective head over an orifice. Then if the jet
issues vertically upwards and H
is not great, it rises to a height
very nearly aqual to S^. It then
expands on all sides (Fig. 38) and
scatters. Let x be the head, measured
from the plane AB, over any cross-
section of the jet, and y the diameter
of the jet at the cross-section. The
velocity of the jet is very nearly
sl'igx and its sectional area is as y^-
But since the discharges at all cross-
sections are equal the velocities are
inversely as the sectional areas,
the jet at the vena contracta where the velocity is JigH^
Fio. 38.
Therefore if d is the diameter of
1 MncyclopcBdia Britannica, ninth edition, Article ■ Hydromechanics.'
52
HYDRAULICS
<!■)'
Or 2/=^(f y ■ • • (27)-
Theoretically ?/ should be infinite when x=Q), but practically the
jet breaks up and scatters. The velocity of the jet decreases
uniformly ; that is, decreases by equal amounts in equal periods
of time. When the head is great the jet does not retain its
coherence long enough to rise to the height H,^
A body of water issuing from an orifice in a direction not
vertical describes, like any other projectile, a curve which, if the
resistance of the air is
neglected, is a parabola
with a vertical axis and
apex upwards. If the jet
issues with velocity V,
and at an angle 6 with
Fia. 39. the horizon (Fig. 39), the
equation to the parabola, as given in Dynamical Treatises, is
y--
f1 ' S6C^
-xt&nd—x"^ — ^^ —
iV"
(28)
where y is the height of any point above the orifice correspond-
ing to any horizontal distance x. The maximum value of )/, that
is the height of the point C above the orifice, is -— sin^6. If ^=0
IV' tan^ V- . ,„«.
= — v)=— sin(26l)
(J sec g
■^9
(29).
This gives the range of the jet on a horizontal plane passing
through the orifice. If ^=45°, a; =
9
This is the maximum range, and in this
V^
case the maximum height is
^9
If the jet issues horizontally (Fig.
40) equation 28 becomes
^=^^2r==4l • • • (-^o)'
and the range of the jot on a horizontal
plane //' feet below the orifice is
x=2^Ejr . . . (31).
The range is a, maximum when H,=H', or, for a plane passing
Pig. 40.
ORIFICES 53
through tho bottom of a reservoir, when the orifice is slightly
below mid-depth. (See also Nozzles, art. 16.)
Section II. — Orifices in Thin "Walls
8. Values of Co-efficient. — The co-efficient c is best known for
circular orifices. It is greater the smaller the orifice. It increases
for small heads. Smith concluded that, with a great head, c was
about '592 for orifices of all sizes. This is disproved by the later
and very careful experiments of Judd and King {Engineering News,
27th Sept. 1906) and Bilton {Min. Proc. Inst. G.E., vol. ckxiv.).
Some of their figures are as follows : —
Diam. of orifice, -75111.(8.) -75 in. (J. & K.) 2 in. (J. &K.) 2-5in. (J.& K.)
Head 4 feet, -613 -609 -608 -596
Head 8 feet, -613 -610 -608 -596
Head 92 feet, -615 -608 -596
Bilton concludes that for each' size of orifice there is a ' critical
head ' H^ which is greater the smaller the orifice and never exceeds
4 feet. For heads greater than- II„ c remains constant. For an
orifice of a given size some observers regularly obtain lower values
of c than others. Any slight rounding of the edge increases c,
especially with a small orifice, and this fact tends to discredit any
specially high figures. But there may be errors in measuring v^
or the diameter of the orifice or the volume discharged. Experi-
ments made in 1898 by Bovey, Farmer, and Strickland give values
of c for "S-inchji 1-inch, and 2-inch orifices generally about -010 less
than those obtained by Bilton and by Judd and King. The causes
of the discrepancies may have been any of those just mentioned.
A complete set of values of c as arrived at by Bilton — and now
accepted — for circular orifices is given in table iv. When the
critical head is reached the co-efficient is underlined. The figures
for heads which are very small, relatively to the size of the orifice,
are not quite reliable. This is chiefly owing to the difficulty of
observing H exactly. Bilton concludes that c is the same in what-
ever direction the jet issues, that it is practically the same for all
circular orifices having diameters of 2-5 inches or more, and that for
smaller orifices it so increases as to become 1 '0 for an indefinitely
small orifice. {Of. table v.)
Barnes ^ arrives at figures which are in excess of Bilton's by
some 1 per cent, for diameters of 1 inch to 2'5 inches, but he does
^ Shown in table vii.
* Hydraulic Flow Reviewed.
54 HYDRAULICS
not take account of Judd and King's figures for the 2'5-inch
orifice. A few experiments sliow that c may continue to decrease
for diameters greater than 2 '5 inches, and figures for three larger
orifices are included in table iv. H^ for these orifices is not
exactly known, but c is practically constant for heads greater than
1'42 feet. For very large orifices in vertical planes Hg must
obviously exceed 1-42 feet. The head 1'42 feet for the three
larger orifices comes within the range of table x., but the values
of c given in table iv. are to be used with the ordinary formula
without correction. (Chap. ii. art. 5, p. 15.)
With square orifices, the streams A and C (Fig. 36) by inter-
fering with one another prevent complete contraction occurring
in the corner. Few experiments have been made, but Smith
concludes that for a square orifice c is about '005 greater than for
a circular orifice of the same diameter and under the same head.
See notes to table iv.
For a triangular orifice c is about "007 greater than for a
square of the same area. This is doubtless because, the angles
being more acute than those of a square, the suppression of con-
traction in them is still greater.
Regarding rectangular orifices other than squares, c can be
compared with that for a square whose side is equal to the short
side of the rectangle. Tables vi. and vii. show values of c arrived
at respectively by Fanning and Bovey. Fanning's figures showing
c as increasing for great heads seem to be slightly inaccurate.
The experiments considered by him did not include heads greater
than 23 feet and only a few of these. The figures in table vi.
above the thick horizontal lines are the uncorrected co-efficients.
Fig. 36 and the text below it show that the jet from a rectangular
orifice is greater, relatively to the size of the orifice, than for a
square, and that the relative size will go on increasing as the
orifice is lengthened. Since, for considerable heads, c is probably
the same for all large square orifices, it would be expected that
for a rectangular orifice c in table vi. would depend only on the
shape of the orifice, i.e. it would be the same for the 4' x 1 ' as for
the r X "25' rectangle. It will be seen that this is not far from
being the case, but that the figures for the greater heads are hardly
in excess of those for the corresponding square orifices. The same
seems to bo the case in table vii. {Cf. variation of figures for
" orifice " in table on p. 68.)
As might bo expected, c is not altered appreciably by turning an
orifice about its axis into a fresh position. See remarks in table vii.
OKIFICES 55
The manner in which c varies for orifices of different sizes and
shapes is the opposite to what it wonld be if the friction of the
oriiice had any appreciable effect. The smaller the orifice, and the
greater its deviation from a circle, the greater is the ratio of the
border to the sectional area, but the greater the co-efficient.
9. Co-efficients of Velocity and Contraction. — The co-efficient
Cy iSj for small heads, about the same for orifices in thin walls as for
bell-mouthed orifices (art. 14). It was found by Judd and King
to be "996 for the smaller orifices and -999 for the larger, the heads
ranging from 7 to 92 feet. It is usual to find c^ by observing the
range of the jet on a horizontal plane (art. 7) — though the resist-
ance of the air may cause some slight error — and to find c,, by
dividing c by c^. Judd and King, however, measured the velocity
of the jet by means of a Pitot tube (chap. viii. art. 14), and they
measured the diameter of the jet at the vena contracta by micro-
meter callipers. The resulting values of c^ and c„ agreed well.
Diameter of orifice = '75 in. 1 in. 1-5 in. 2 in. 2-5 in.
Cc =-613 -612 -605 -608 -596
The distance from the plane of the orifice to the point where the
jet attained its minimum section was 'SD for the ■75-inch and 'SD
for the 2'5-inch orifice (D being the diameter of the orifice), and the
jet thereafter continued to have th6 same section. Bazin found
the section, after the vena contracta had been passed, to continue
to contract, but very slightly.
10. Co-efficients for Submerged Orifices. — All the co-efficients
above mentioned are for cases in which the orifice discharges into
air. Table viii. shows the results found by Smith for drowned
orifices, the downstream* water being -57 feet to '73 feet above the
centre of the orifice. The co-efficients are less by about 1 per cent. ,
or for small sizes 3 per cent., than for similar orifices discharging
into air. The cause may perhaps be the formation of eddies, and
the friction of the jet against the water surrounding it.
The following co-efficients (C) were obtained by Stewart.^ The
tubes had sharp upstream edges. They were of wood and fixed in
a 10-foot channel, with margin at each side 3 ft., at bottom 2'9 ft.,
at top about 2 ft. j (average) was only 'GS ft., so that the con-
traction was not complete. It was wholly suppressed on one or
more sides, as noted in column 1 of the table, by adding curved
approaches.
11. Remarks. — If an orifice in a thin wall is in a surface not
^ Engineering News, 9tb Jan. 1908,
56
IIYDEAULICS
SUHMBRGED OrIFIOBS AND TuBBS 4 FeeT SqUARB.
Suppres-
H
Length of Tube in
Ft. and Class of Orifice.
sions,
ill
■31
■C'Z
Vio
2-B
6
10
14
k
la.
Thin
■Wall.
Intermediate.
Cylindrical Tube.
■05
63
■65
•67
•77
•81
■82
■85
■10
•61
■63
•65
•72
•76
■78
•80
Nil -
■20
•61
•63
•65
•71
■77
■79
•81
■25
•61
•63
•65
■78
•81
•83
.
■30
■61
•64
•66
•80
•83
■85
r
•05
■67
•74
•81
•85
Bot-
tom
■10
■64
•70
•77
•80
•20
•25
•63
■63
•69
•78
•79
•82
.
•30
•64
Bot- ■
•05
•74
•77
■83
•86
torn
■10
•69
•72
■79
•81
and
■20
•68
•71
■80
•83
one
•25
•68
■81
side
■30
•69
Bot- '
■05
•83
•77
■88
•89
torn
•10
■77
•72
■83
•84
and -
■20
•77
•71
■84
•86
two
•25
•78
•85
sides
•30
•79
,
•05
•95
■94
•94
•93
•93
All
•10
•93
•91
■90
■89
•89
four -
■20
•96
•92
•91
•91
•91
sides
■25
•30
•97
■98
•93
•
In this group A = 9'ia' and
e — C. In other groups c < C.
In final group A = 56a and c
some 2'5 per cent < C.
In every column C reaches a
minimum value as B increases.
It increases again when H is fur-
ther increased. Similarly with
other groups.
For the 2-5-foot tube the sup-
pressions produce no effect
Pressure of air surrounding jet
(art. 12) probably increased.
^^( Values of C for the 14-foot
■83 )- tube when a cross bulkhead
g5^ was added at tail end.
Ordinarily no tail bulldiead
existed, and bacl£ eddies formed
along sides of tube.
It is only in this group that
suppression of contraction much
affects the cylindrical tubes.
For cylindrical tubes in gen-
eral see art. 12.
plane, the co-efficient -will be greater or less than for a plane
surface, according as the surface is concave or convex towards the
reservoir.
In some districts in America, -where •water is sold for mining
purposes, the quantity taken is measured by orifices. The
' Miner's Inch ' is a term ■which often means the quantity of water
discharged by an orifice 1 inch square, in a \ertical thin wall,
under a head of 6| inches. In this case, if r is taken at -621, Q is
1 '53 c. ft. per minute ; but the head is not always the same, and
the orifices used arc of many different sizes, generally much larger
than a square inch ; the Miner's Inch is then some fraction of the
total discharge, and its value in c. ft. per minute varies from 1-20
to 1-76. The Minor's Inch is, in fact, a name with local varieties
ORIFICES
57
of meaning. The wall containing the orifice is often made of
2-inch plank, and the chief practical point to be noted is, that
with a small orifice, or a very long orifice of small height, not
only is exactness of size more difficult to attain, but there may be
a chance of the orifice acting as a cylindrical tube, and giving a
greater discharge than intended. Before the discharge of the
orifice can be known, the size, shape, head, degree of sharpness,
thickness of wall, width of clear margin, and velocity of approach
must all be known.
=F==zi&^
._i
Section III. — Short Tubes
12. Cylindrical Tubes. — In a cylindrical tube (Fig. 41) the jet
contracts, but it expands again, fills the tube, and issues 'full
bore.' The sectional
area at GK is, as in a
simple orifice in a thin
wall, about -GS times
the area at LM, but the
velocity at OK is
greater than iJigH,
and the discharge
through the tube is
greater than that from
an orifice of area LM.
When the flow first
begins, the air in the
spaces NG, KO is at
the atmospheric pres-
sure, and the discharge
is not greater than that
from an orifice LM.
The action of the water
exhausts the air and produces a partial vacuum. Let p be the
pressure in NG, KO. The pressure in the jet GK is also^. The
pressures at QB and ST are P„. Let V, v be the velocities at GK
and QB. The loss of head from shock between GK and QB
^ '-. Then from equation 5, p. 11, if
Fig. 41.
(equation 18, p. 32) is
the tube is horizontal,
(A)
58 HYDEAULICS
But «;=-63rand F-v=-37F.
Therefore from (B) H=:~'- [(■63)^ + (-37)4=^5^i^
Practically there is some loss of head between LM and GK, and
actually
F=:l-30j2gl{ ._^. (32),
«;=-63r='82v/2^ir . . . (33).
Also from (A) £'+^=|,+|!
=|,+ (l-30)^^.
Therefore ^- 1^= -egif . . . (34),
Or the pressure at GK is less than the atmospheric pressure by
'69JVII. The result is nearly the same if the tube is not hori-
zontal, provided 3 is large relatively to the length of the tube.
If c„ is not exactly -63, or if the actual loss of head differs from
that assumed, the above results are somewhat altered. With a
great head the vacuum becomes more perfect, the contraction,
owing to the diminished pressure on the jet, less complete, and
the figures I'SO and '69 are reduced. For moderate heads they
are found to be about 1'32 and -75.
If holes are made at N, 0, water does not flow out but air
enters, and the discharge of the tube is reduced. If a sufficient
number of holes are made, or if the whole tube and reservoir are
in a vacuum, or if the tube is greased inside, so that water cannot
adhere to it, the discharge is no greater than for a simple orifice.
If the holes are made at a greater distance from LM than about
IJ diameters the discharge is unaffected. If a tube is added
communicating with a reservoir E, the water for ordinary heads
rises to a height EF=-75H, and if the height EO is less than
this, water will be drawn up the tube and discharged with the jet.
This is the crudest form of the 'jet pump.' The height to which
water can be pumped, even if the vacuum is perfect, is limited to
34 feet. The discharge of the tube is reduced by the pumping.
With a great head the quantity -IbH may exceed 34 feet, but
in no case can the difference of pressures exceed that due to
34 feet.
ORIFICES
59
The co-efficient of discharge for a cylindrical tube, like that for
a simple orifice, increases as the head and diameter decrease. The
approximate values are given in table ix., but the number of
observations made has not been great. For large tubes see p. bo.
CD
The co-efficient for a tube AGO or AGQB
(Fig. 42), CD being ABx -79, has been found to
be the same as for a simple cylinder.
13. Special forms of Cylindrical Tubes. p,o. 42.
—If the tube projects inwards (Fig. 43) the contraction and loss
of head by shock are greater than in the preceding case, and if
the edge of the tube is sharp the co-efficients c, and c are reduced
to about '72. This is because some of the water comes from the
directions AB and CD-. For small tubes see table v.
When the length AG (Fig. 44) is so short that the jet does not
again touch the tube, it is known as Borda's mouthpiece. For
Fm. 43. Fig. 44.
small heads AG is about half of AB. The co-efficient c„ is about
the same as for a simple orifice, but the contraction is greater. It
is the greatest that can be obtained by any means. The value of
Cc is -52 to '54. That of c is '51 to '53, and it does not vary much.
The jet also retains its coherence longer than those from other
kinds of orifices.
The co-efficient for Borda's mouthpiece can be found theoreti-
cally. The velocity of the fluid along the sides of the reservoir
FD, SG, which in most orifices is considerable, is here negligible.
Thus the pressures on all parts of the reservoir are taken to
be the simple hydrostatic pressures, and they all balance one
another except the pressure on GS, which, resolved horizontally, is
Wa(H+^^. The horizontal pressure on AMNB is P„«. The
difference between the two is WaE. In a short time t let the
60
HYDKAULICS
■water between KL and MN come to 8TQP. Its change of
horizontal momentum is the difference between the horizontal
momenta of KSTL and of MNQP, and that is the horizontal
momentum of MNQP, since KSTL has no horizontal momentum.
This change of momentum is caused by the force WaH. Equating
the impulse and momentum,
miHt=JVQt-=
9
9
■■WcriVt^.
Therefore
Let
Then
Or
When a tube is placed obliquely to the side of the reservoir (Fig. 45) the
co-efficient ia about c-'0016S where B is the number of degrees in the
angle made by the axis of the tube with a line perpendicular to the side of
the reservoir, and c is the co-efficient for the tube when 9 is 90° (Neville).
Fia. 46.
Pig. 46.
For a cylinder with a thin diaphragm at its entrance (Fro. 46) the following
co-efficients are given by Neville. They apply only ^\-hen the tube is filled,
which it will be if not too long nor too short.
Ratio o( Area.
Co-efflcient of Dischareo '
AB to area CD.
for CD. 1
•0
•000
•1
•066
•2
•139
■3
•219
•4
•307
•5
•399
■6
•493
■7
•587
■8
•675
•9
•753
1-0
•821
ORIFICES
81
•61
1'64
11-48
55-77 337^93
959
■967
•975
•994 -994
14. Bell-moutlied Tubes.— A simple bell-mouthed tube (Fig. 8,
page 12) is made of the shape of the jet issuing from an orifice
in a thin wall. The length BE is half the diameter AB, and the
curves AG, BD have a radius of TSO times AB. This makes
CD= •SO X AB. The edges at A and B must be rounded and not
left sharp. Weisbach found the following co-efficients for small
bell-mouthed tubes : —
Head in feet :
Co-efficients (c„ and c) :
This form of tube is often used as a mouthpiece for pipes to
prevent loss of head by contraction. If the tube is not carefully
made according to the above description c will probably not
exceed ^95. For tubes of square cross-section 1 foot in
diameter resembling bell-mouths co-efficients of -94 and ^95 have
been found.
15. Conical Converging Tubes. — In a conical converging tube
(Fig. 47) the stream contracts on entering and again on leaving
the tube. The co-efficients vary with
the angle of the cone, but c^, is always
greater than for a cylinder. The follow-
ing table shows the co-efficients found by
Castel for a tube whose smaller diameter
was •61 inch, and its length 2^6 times
the smaller diameter. The co-eificients
have reference to the smaller end of the
tube. As the angle of the cone increases
Cj diminishes and c^ increases. Their
product c is a maximum for an angle of
13° 24'. The co-efficients were found to
be independent of the head. piq. 47.
Angle of cone = 0° 0'
l" SB'
4° 10'
r 52>
10° 20'
13° 24'
16° 36'
21° 0'
29° 5S'
40° 20'
48" 50'
cc = 1-000
1-OOS,
1002
•998
•987
•983
•969
•945
•919
•8S7
•861
Cv= -330
■866*
•910
.931
•960
•962
•971
•971
•975
•980
•984
c = -829
•866
•912
•929
•938
■946
•938
•918
•896
•869
•847
If the angles at the entrance are rounded off so as to form a
bell-mouth, c is increased by about -015.
62
HYDRAULICS
The following have also been found : —
Cross-
section
of Tube.
Head
in
Feet.
Smaller end
of Tube.
Larger end of
Tube.
Length
of Tube.
Angle of
Conver-
geuce.
..
Circle
300
1'20 in. diam.
4 '20 in. diam.
10 ins.
17'
1-00
Circle
2-7
1 21 „ »
1-50 „ „
[275 „ „ \
•92 „
4° 20'
•9.34
•903
Circle
1-8
217 „ „
13-50 „ „ 1
ISO „ „ \
[9-83 „ „
7-67 „
10'
20°
45°
•898
•888
•864
Rect-
9-6
•44 ft. X -62 ft.
2-4 ft. X 3-2 ft.
9-59 ft.
11° 38'
■976
angle
and
15° 18'
to
•987
Conical converging tubes are used to obtain a high velocity,
but the above tables show that the velocity is not generally
greater than for a bell-mouthed tube. The angle is usually 10°
to 20°. A cylindrical tip is sometimes added, its length being
about 2J times its diameter. In the case shown above, with a
head of 300 feet, the jet did not touch the cylinder. If the tube
projects inwards into the reser-
voir the co-efficient is reduced,
but is greater than for an in-
wardly projecting cylinder. Coni-
K-5-3
cal tubes (Fig. 48) are used in
India at canal falls for delivering
streams of water on to wheels for
^^°- ^^- driving mill-stones. There is loss
of head both at the entrance and at the bend. The loss would be
reduced by using a bell-mouth and a curve.
16. Nozzles. — In order to give a high velocity to the stream
Fia 49.
Fig, 60.
X
^
Pia. Bl. Pio. 62.
issuing from a hose-pipe a nozzle is applied to its extremity.
Figs. 49 and 50 show 'smooth nozzles,' and Figs. 51 and 52
ORIFICES
63
two forms of 'ring nozzle.' The diameter, d, of the orifice is
usually about one-third of the diameter, D, of the pipe, and the
length of the nozzle six to ten times d. Experiments with nozzles
have been made by Ellis, Freeman, and others.^ The pressure, p,
at the entrance to the nozzle being measured by a pressure-gauge,
The following co-efficients have
the head on the nozzle is 4=,
W
been found for the smooth nozzles, the pressure being 15 to
80 lbs. per square inch.
Diameter of orifice = | in. f in. 1 in. li in. \\ in.
f„ =-983 -982 -976 -972 -971
For the ring nozzle c is for Fig. 51 about '74, and for Fig. 52,
where a Borda's mouthpiece is added, about '52. In both cases
c„ is about the same as for smooth nozzles.
D AD"
To allowforvelocityofapproach,since-=-= 3, therefore —=-=^ — Q'0.
From table ii., noting that c^ is greater than -97, it is clear that c„
is about I'Ol, and the true co-eificient c must be increased 1 per
cent, to give the inclusive co-efficient C.
The following table shows the vertical heights attained by jets
from nozzles in experiments made by Ellis. It will be seen that
the height of the jet is greater for the smooth nozzle than for the
ring. It is also greater the larger the diameter of the nozzle, and
this may be due to the jet longer retaining its coherence.
Vertical Heights of Jets from Nozzles.
Pressure
in pounds
per square
inch.
Pressure-
head in
feet.
l-inch Nozzle.
l:i-inch Nozzle.
5-inch
Nozzle.
Smooth.
Ring.
Smooth.
Ring.
Smooth.
10
20
30
50
70
100
23
46
69
115
161
230
22
43
62
94
121
148
22
42
61
92
115
136
23
43
63
99
129
164
22
43
63
95
123
155
59
92
113
133
The total height to which the jet remains serviceable as a
fire-stream is less than that to which the scattered drops rise, the
former height being about 80 per cent, of the latter for small
^ Transactions American Society of Civil Engineers, vol. xxi.
64
HYDRAULICS
heads and GO or 70 per cent, for greater heads, but it is difficult
to say exactly to what height the stream is serviceable. The
heights given in the above table are the total heights. Many
kinds of nozzles have been tried, but with none of them does the
stream remain clear, polished, and free from spraying up to the end
of the first quarter of its course. Such a stream can be obtained
for a pressure of 5 or 10 lbs. per square inch, but not for a good
working pressure.
17. Diverging Tubes. — With a conical diverging tube (Fig. 53)
the jet contracts on entering and expands again. With a tube
having an angle of 5°, smaller diameter
1 inch, and length 3| inches, the co-
efficient of discharge for the smaller end
was -948 ; but with a tube having an
angle of 5° 6' and a length of nine times
the smaller diameter, a co-efficient of
1-46 was found. The case is similar to
a cylindrical tube. If the angle exceeds
7° or 8° the jet may not fill the tube,
and the co-efficient is then reduced. If
the angle is further increased, the jet
does not touch the tube, and the case
becomes an orifice in a thin wall.
If the tube projects inwards into the reservoir the co-efficient is reduced,
but is greater than for an inwardly projecting cylinder. If the length of
the tube is now reduced so that the jet does not touch the tube, the co-
efficient is greater than 'SI, the value for Borda's mouthpiece, and becomes
about '61 if the taper is increased till the case becomes a simple orifice.
A compound diverging tube (Figs. 54 to 60) consists of a
converging or bell-mouthed tube with an additional length in
which the tube expands again. If there are no angularities no
head is lost by shock. The case is similar to that of a cylindrical
tube. The pressure at the discharging end of the tube being Fa,
the pressure at the neck is less because of the higher velocity.
The following table contains information regarding various
diverging tubes. It is clear that the co-efficient increases with
the ratio of expansion (column 5) and decreases as the taper
(column 6) increases, the highest co-efficients being obtained with
high ratios of expansion and gentle taper. With a mean taper
of 1 in 13"7 the limit seems to be reached when the ratio of
expansion is 3'15, but with a taper of 1 in 5'33, not till the ratio
is 5-0.
OEIFICBS 65
A negative pressure in the neck is impossible (chap. ii. art. 1),
but if the vacuum there were perfect the pressure would be zero
and the velocity would be U ig(li+^^ or V2(/(if+34). By-
making H small the discharge could be increased enotmously, but
practically the vacuum is always imperfect, and at a certain point
the water ceases to fill the tube at the discharging end. The
maximum co-efficient ever obtained is 2 '43.
The remarks regarding pumping action made under cylindrical
tubes apply equally to diverging tubes. In a vacuum or with a
greased tube the discharge from a diverging tube is no greater
than from the mouthpiece alone, and the same may be the case
with a great head, the stream passing the expanding portion
without touching it.
66
HYDRAULICS
FlQ. 54.
Pio. 65.
Fia. 56.
In Figs. 54, 55, and 56 ^5 = 1-5 in., CD = l-2l in., AC= -92 in.
— D C ■R.A
Fig. 67.
Fio. 59.
In Pigs. 57, 58, and 5'd AB is a bell-mouthed tube with diameter at B = | in.
All the other segments except DE (Fig. 57) are conical, and each is 2 m.
long.
Fin. 60.
In Fig. 60 the piece AS has a cyoloidal curve and BP is cylindrical. The
other pieces, each 1 ft. long, are cnnical, but the angle of the cone is least for
PQ and increases for each successive piece.
The tubes were submerged. The head varied from -1 ft. to I'Sft., the
co-eflBcient generally increased with the head (probably because the vacuum
was more complete), the values 2'08 and 2'43 with the tube AS being for
heads of '13 ft. and 1 '36 ft. respectively. But for a head of 1 -39 ft. the co-
efficient was 2 '26.
ORIFICES
67
(1)
(2)
(3)
, <*>
(5)
(6)
Ratio of
Reference
Co-efflcient
Smallest
Diameter at
Discharging
End to
Taper of Tube, or
to
Tube.
for Smallest
Dia-
Rate at which
Figure.
Diameter.
meter.
Diameter increases.
Diameter.
Inches.
Fig. 54
AE
1-40
1-21
2-48
1 in 5-5
„ 55
AE
1-38
1-21
1^24
1 in 14-1
„ 55
AG+C'E
1-43
1-21
1^24
1 in 14-1
„ 56
AE
1-57
1-21
1-59
1 in 9^1
Fig. 57
AG
1-52
■375
1^58
1 in 9-1
AD
1-78
•375
2-17
1 in 9-1
AE
1-87
•375
3-83
1 in 5^6 (mean)
„ 58
AE
1-69
•375
2^33
1 in 4^0
AG
1-79
•375
3-67
1 in 40
AS
1-79
•375
3-33
1 in 6^6 (mean)
„ 59
AK
1-88
■375
2-0
1 in 533
AL
203
•375
3-0
1 in 5-33
AM
2-07
•375
4^0
1 in 5-33
AN
2 09
■375
5-0
1 in 5-33
AO
2-09
■375
6-0
1 in 5^33
Fig. 60
AQ
a -48 to 1-60
1-22
r42
1 in 23-3
AR
1-98 to 2-16
1-22
2-30
1 in 15'1 (mean)
AS
2-08 to 2-43
1-22
3^15
1 in 13'7 (mean)
AT
2-05 to 2-39
1-22
4-0
1 in ]3'1 (mean)
In Fig. 54 ^^=3 in.
In Fig. 55 ^i?'=1^5in.
C'J'=9-75 in.
C(7' = 3-Oin.
OD = C'D'
G'E = i-l in.
In Fig. 56 EF=l-92 in. GE =6-5 in.
In Fig. 57 Diameters at C, D, E are J| in., if in., 1^ in.
In Fig. 58 Diameters at F, O, H are \ in., 1§ in., \\ in.
In Fig. 59 Diameters at K, L, M, N, are | in., 1^ in., Ij in.. If in.,
2iin.
In Fig. 60 Diameters at B, P are 1^22 in., and at Q, R, S, T 1^74 in.,
2-81 in., 3^85 in., 4-90 in.
68
HYDRAULICS
Co-efficients
FOR Sluices, etc.
Kinds of
Description.
Width of Height of
Co-efficient.
Head.
Aperture.
Opening.
Opening.
Shown in Fig. 61.
2'0 ft.
1-31 ft.
■61 to -m
■33 ft. to
to -10 ft.
(averages)
9^8 ft. over
upper edge.
Sluice,! . ]
■64 to ^70
Do.
Aa above, but with
Do.
Do.
(averages)
boards CF or DE
added.
1-7 ft.
Do.,
In woodwork 1'77
4-265
■625
6 ft. to 14
ft. thick at b ottom ,
ft.
ft. over
and -87 ft. else-
Do.
■39 ft.
■803
centre.
where.
Iron gates,^
Working in grooves
4 ft. to
3 ft. to
■72 to -78
•25 ft. to
Bari Doab
in the masonry
10 ft.
2 ft.
(averages)
4-8 ft.
Canal,
heads of distribu-
India.
taries.
Orifice,'
Shown in Fig. 62.
■5 ft.
•5 ft.
■593
■5 ft. over
1-inch plank placed
10 ft.
>»
■607
upper edge.
against a 6-ineh
1-5 ft.
,,
■615
space between two
2 ft.
jj
•621
2-iuch planks.
2-5 ft. ' „
■626
I'— ^ :
to
* The smaller values of c oc-
curred with the greater height of
opening. For any given height of
opening c varied as the head
changed, being generally greatest
for a head of about 1 ft.
^ The co-efficient includes the
allowance for velocity of approach,
which was considerable. There was
no contraction at the bottom and
aides. The openings were gener-
ally submerged. C increases as H
decreases, and it also increases with
the size of the opening.
' The co-efficient varies in a
similar manner to that for an
orifice in a thin wall.
Fio. 81
ORIFICES
69
Fio. 62.
Section IV. — Special Cases
18. Sluices and other Apertures. — A sluice is an orifice pro-
vided with a gate or shutter. Generally there are adjuncts which
complicate the case and render the co-efficient uncertain. When
the gate is fully open the case may approximate to that of an
orifice in a thin wall. When it is nearly closed the case may
resemble that of a prismatic tube. Where accuracy is required
the co-efficient must be determined experimentally. It may have
any value from '50 to '80, or even outside these limits. The
preceding table shows some values. Sometimes when a thick gate
is lifted the flow tends to force it down again, especially when it
is raised slightly. This is
probably due to the forma-
tion of a partial vacuum
under the gate.
If the sides and lower
edge of an orifice are pro-
duced externally so as to
form a 'shoot' (Fig. 63)
the co-efficient c may be
greatly altered. The air has
access to the issuing stream,
so that reduction of pres- ^ ^^
■^ Fig. 63.
sure in the vena contracta
cannot take place, as in a cylindrical tube. On the other hand the
^- ^//a//j//m/^////M/////////MMmv7M/m/»Mmmw/MMm
70
HYDRAULICS
friction of the shoot has to be overcome. When the head is more
than two or three times the height AB the discharge of the shoot
may be nearly the same as that of the simple orifice, but otherwise
it is reduced. For an orifice 8 inches by 8 inches with Hi 4|
inches the addition of a horizontal shoot 21 inches long reduced
c from -57 to "dS. With a horizontal shoot 10 feet long the
following co-efficients have been found,^ the orifice being '656
feet wide, if, and H^ are the heads over the upper and lower
edges of the orifice.
Ih-«1-
Hi in feet.
Remarks.
■066
•164
•328
•656
l-Bi
9 •81
feet.
■656
■164
■48
■49
•51
•58
■54
•62
•57
■63
•60
•63
•60
•61
[-Full contraction.
\Lower edge of orifice flush
J with bottom of reservoir.
1
■656
■164
■53
■59
■55
■61
■57
•63
■59
■65
•61
■65
■61
■65
19. Vertical Orifices -with small Heads.— Let ACDB (Fig. 64)
be a bell mouthed orifice.
The equations for orifices of
diflferent forms are found by
integration. An orifice is sup-
posed to be divided into an
infinite number of horizontal
layers. The discharge of any
layer is c.„JigH-ldH where H
is the head over the layer, I its
length in the plane of the orifice,
and dH its thickness. For a
rectangular orifice
Q=c,jJ-2ff/ 'l
J H,
PlO. 04.
=lcjj2g{H,^-H,^). . . (35),
where H, and H^ are the heads
at (' and D respectively. The
discharge is the difierence between the discharges of two weirs
' Morin'a J [ i/drauliqiic, second edition, pp. 36 and 37.
ORIFICES "ri
with crests at C and D respectively, and no contraction. For a
triangle whose base is upward and horizontal and of length I
Q = l.cjj2glr^^-II.^ . . . (36).
For the same triangle with base downwards and horizontal
Q=.^cJ^2g(s,i-f^^^'j . . . (37).
For a trapezoidal orifice, the lengths of whose upper and lower
sides are l, and l^ respectively, these sides being horizontal, the
equation is obtained from equation 35 with 36 or 37. It is
Q=§c,^2gi^kHfi-W,i+Uk-k)^^-:r^'^) . . . (38).
For a circle whose radius is B and H the head over its centre
If velocity of- approach has to be allowed for nh must be added
to each of the heads in equations 35 to 39. Thus equation 35
becomes
Q^%cjj2~g{{H,+nh)i~{Ht+nhf} . . . (40).
In every case the discharge calculated by the above equations is
less than that obtained with the same co-efficient by equation 9 or
10, p. 14, but owing to the much greater simplicity of these last,
it is better to use them, and to multiply the result by a second
co-efficient to correct the error. These 'co-efficients of correction,'
c^, are given in table x.^ In this table D is the height, measured
vertically, between the upper and lower edges of the orifice and
D (Fig. 64), and the head in column 2 is that over a point halfway
between these edges. This, in the case of triangular or semi-
circular orifices, is not the head over the centre of gravity of the
orifice,^ but this latter head must be used in equation 9 or 10.
The correction required is practically negligible when H— 2D. It
is greatest when H='5QD, that is when the upper edge of the
orifice is at the surface, which of course it never can be exactly.
All the above equations apply to orifices with sharp edges, but
they ought to be applied to the vena contraota. Not only is
D less for CD (Fig. 34, p. 45) than for JB, but H is greater
because of the fall FN. which the jet undergoes between AB and
CD. Thus the ratio in column 2 of table x. is always greater for
^ Smith's Hydraulics, chap. ii.
^ The distance of the centre of gravity of a semicircle from its diameter is
•4244 of the radius.
72 HYDKAULICS
CD than for AB. The coefficients for orificeia in thin walls, those
which are above the horizontal lines in the columns of table vi., have
however been obtained by applying the above equations to the
orifice AB, and for such orifices the co-efficients should be so used, or
if equation 9 or 10 is used, c^ should be taken with reference to AB.
But for a sluice, cylindrical tube, or other aperture for which some
other co-efficient c is to be employed, the correct method is to
ascertain c„ and c„, obtain the approximate dimensions of the jet, and
find the fall PNhy equation 31 (p. 52). This has been done for some
square orifices, and the results utilised by adding column 1 to
table X. For any entry in this column the corresponding entry in
column 2 gives the approximate figure for the jet, and the value
of Cj (to be applied to the result found by equation 9 or 10) is that
in column 3. For a rectangle whose horizontal side I is less than
I), the vena contracta is nearer to the orifice, the fall PN is less,
and the contraction of the jet in a vertical direction less, so that
the figures in column 1 approach nearer to those in column 2.
When I is less than SD column 1 is not needed.
The co-efficients for vertical orifices under small heads are not
well determined. The smallness of the margin on the upper side
of the orifice tends to produce incomplete contraction there and
to increase c ; but, on the other hand, there is a fall in the water-
surface upstream of the orifice, the head is measured above the
fall, and this, according to Smith, reduces c. A vortex may also
be formed, and possibly it may penetrate the orifice and reduce
c. For the above reasons the corrections are of use chiefly for large
orifices. They could, for instance, be applied to Stewart's co-efficients
(page 56) for cases of free — not submerged — discharge.
With an orifice in a horizontal plane under a small head the
proportion of water approaching axially is reduced and the con-
traction is probably increased, except with bell-mouths. The
co-efficients for such cases having nearl}- all been obtained for
orifices in vertical planes, are not likely to apply correctly to
others, even if the head is measured to the vena contracta.
The matter in this article refers to cases where H is small
compared to the orifice. If, in addition, 11 is actually small, the
difficulties attending such cases (chap. ii. art. 7) are added.
Examples
Example 1. — Water enters the condenser of a steam-engine at
the soa-lcvcl from a reservoir whose A\'atcr-surface is 10 feet above
the injection orifice. The pressure in the condenser is 3 lbs. per
ORIFICES 73
square inch. Find the theoretical velocity of flow into the
condenser.
The atmospheric pressure in the reservoir is 14:'7 lbs. per square
inch. The resultant pressure is thus 11 '7 lbs. per square inch or
1685 lbs. per square foot. This is equivalent to a head of
/r7r-r = 27 feet. The total effective head is therefore 37 feet.
From table i. the velocity is 48 '7 feet.
Example 2. — Find the discharge from a circular bell-mouthed
tube, 1 foot in diameter, situated in the middle of the end of a
horizontal trough of rectangular section, 2 feet wide and 2 feet
deep.
The head is 1 foot. From the table in article 14 c,„ is probably
•96. From table x. the co-efficient of correction for small heads
is '992. ^ is 4 square feet and a! is •7854 square feet.
A 4
~= noat =5'01. From table iii. the co-efficient of correction for
a ^7854
velocity of approach is 1'02. From table i. \/2gII=8-02. Then
Q= -96 X 8^02 X -785 X -992x1 -02 = 6^1 2 cubic feet per second.
Example 3. — A culvert 3 feet long, consisting of a semicircular
"arch of 1 foot radius resting on a level floor, has to pass a
discharge of 9 eft. per second. There is a free fall downstream.
What will be the water-level upstream 1
From table ix. c may be taken to be -80. Also a=2x '785 = 1-57
square feet.
To obtain an approximate solution
(^=9=-80 V27f/xl-57 .-. j2gE=r^-Q^j:^=7-l7.
From table i. .iZ"= -80, or the water will be -80 foot above the
centre of gravity of the aperture or -22 foot above the crown of
the arch.
The contraction, supposed to be complete elsewhere, is nearly
absent at the crown, and may be taken to be suppressed on one
fourth of the perimeter, thus (table ii.) making
c=-80xl-04=-832.
In table x. D = 1'0 foot, and the head over the centre of the
orifice is •22 + -50=-72 foot or -IW. This corresponds to -SQD for
the vena contracta, and the figure in column 8, differing, no doubt,
hardly at all from column 4, is -989.
The above two corrections are 4 per cent, plus and 1 per cent,
minus, so that Q is really 3 per cent, more than assumed. To
make it right deduct 6 per cent, from H, which will thus be
■80 X -94 = -752 foot, that is, the water is "18 foot above the crown.
74 HYDRAULICS
Example 4. — For the culvert shown in the annexed diagram
(2 feet wide and 5 feet long), let there be an
open approach channel 4 feet wide, with vertical
walls and floor level with that of the culvert.
Find the discharge when the upstream head
is 1 foot above the crown of the arch, and the
downstream head 6 inches above it.
In this case there is incomplete con-
traction on all sides, and also velocity of
approach. From example 3, a=3-57 square
feet; v^ = 12-0 square feet; P=4-0+3-14 = 7-14 feet; <S'=2-0feet.
If the contraction were complete on AEB, c^ would be (art. 3)
about •80x(l + -152xf) = -80xl-043=-834. The average margin
G 1-30
on AEB is about 1-30 feet. Therefore ^~~2~~^^' ^^^
p = «r^=-75. From table ii. -=1 '035 about. Therefore
c~ -834x1 -035 = -863.
The head is -5 foot, and as the orifice is wholly submerged no
correction for small head is needed. From table i. ■j2gH is 5*67.
$=-863x5-67x3'57 = 17-47 cubic feet per second.
To allow for velocity of approach by the usual method,
17-47
«= ■■ ■ = 1 -46 feet per second. Let «,= 1 -0.
From table i. A=-033, ir+A=-533. From table i. r=5-87.
Then Q= -863 X 5-87 x 3-57 = 18-08 cubic feet per second.
To allow for velocity of approach by a co-efficient of correc-
tion, for the contracted section c„ is (art. 12) about 1-30, and
c„= y^hq =-664. Therefore a' = 3-57x -66 = 2-36 square feet, and
A 12-0
— = ~7o^ = 5-09. From table iii., noting that c, is about 1-30
instead of -97, and that the figures in column 3 are to be increased,
c„ is about 1-03, that is, 3 per cent, must be added to 17-47,
making 17-99 cubic feet per second.
Note. — Further exam])les may be obtained by taking cases analogous to
some of those in examples of chap. iv.
Table I. — Heads and Theoretical Velocities. (Art. 1.)
For a head greater than 10 feet divide the head by 100 and
take ten times the corresponding velocity. Thus for a head of
ORIFICES
75
120 feet the velocity is 87 "9, or ten times the velocity given for a
head of 1-2 feet. For a velocity over 25 divide it by 10 and
multiply the corresponding head by 100. The same methods can
be adopted to facilitate interpolations. Thus for H=-032 look
out 3-2.
In the first fifteen entries the heads correspond to certain
definite velocities. These entries may be useful in cases of velocity
of approach. After that the velocities correspond to definite
heads.
H
r
H
V
H
r
H
V
H
r
H
F
•0022
•38
•13
2-89
■45
5^38
■84
1-35
2^3
122
6-2
200
•0025
■40
•135
2-95
•46
5^44
•85
7-40
2^4
12^4
63
201
•0027
■42
•14
3-00
•47
5-50
■86
7-44
2-5
12^7
64
203
•0030
•44
•145
3-05
•48
5^56
■87
7^48
26
12^9
6-5
20-5
•0033
•46
•15
311
•49
5-62
■88
7-53
2-7
13-2
6 6
20^0
•0036
•48
•155
316
•50
5-67
•89
7^57
2-8
134
6^7
20^7
•0039
•50
•16
3-21
•51
5-73
•90
7^61
2-9
13^7
6^8
20-9
•0042
•52
•165
3-26
■52
5-79
•91
7-65
3
139
6-9
2ro
•0045
•54
•17
331
•53
5-85
•92
7^70
31
141
7
21-2
•0049
•56
•175
3^36
■54
5^90
•93
7^74
3-2
14^3
7^1
213
•0052
•58
•18
3-40
■55
5^95
•94
7^78
33
14-5
7-2
21-5
■0056
•60
•185
345
■56
600
•95
7^82
3^4
14-8
73
21-6
•0066
•65
•19
350
•57
606
•96
7^86
35
15^0
7.4
21-8
•0076
•70
•195
3-55
•58
611
•97
7^90
36
15^2
7^5
21^9
■0087
•75
•20
359
•59
6^17
•98
7-94
37
15^4
7^6
221
■01
•80
•21
3^68
•60
6^22
■99
7^98
3^8
15-6
7^7
222
•015
•98
•22
376
•61
6^28
1
8-02
3 9
15^8
7^8
22^4
•02
113
•23
3-85
•62
6^32
1^05
8-22
4
160
7-9
22^5
■025
1^27
•24
393
•63
6-37
M
8^41
41
162
8
22-7
•03
139
•25
4^01
•64
642
M5
8^eo
4-2
16^4
8^1
228
035
1-50
•26
4-09
•65
6-47
1^2
8^79
4-3
166
8^2
230
■04
1-60
•27
4^17
■66
652
1^25
8-97
4-4
16^8
83
231
•045
i-io
•28
4^25
■67
6^57
13
9-15
4^5
17^0
8-4
23^2
■05
1^79
•29
4^32
■68
6^61
135
932,
4-'6
17^2
8^5
234
■055
1^88
■30
439
•69
606
r4
9^49
4^7
17^4
8^6
23^5
■06
1-91
■31
4^47
•70
671
1-45
9-66
4-8
176
8^7
23-6
065
2-04
■32
4^54
•71
6^76
1^5
983
4^9
17^7
8-8
238
•07
2-12
■33
4^61
•72
681
r55
998
5
17^9
8^9
239
•075
2^20
■34
4-68
•73
6-86
16
10^2
5^1
181
9
241
■08
2-27
■35
■36
4^75
•74
691
1-65
10-3
52
18-3
91
242
■085
2^34
4^81
■75
6^95
1-7
10-5
5^3
18^5
9^2
24^3
■09
2^41
■37
4-87
■76
6-99
1-75
10^6
5^4
18-7
9-3
244
■095
2^47
■38
4^94
■77
7^04
1^8
10^8
5-5
18-8
9^4
24-6
•10
254
■39
501
■78
7-09
1^85
10-9
5-6
19
95
24^7
■105
2-60
■40
5-07
■79
7^13
1-9
IM
5^7
19-2
9-6
24^8
•11
2-66
■41
5-14
■80
7^18
7^22
r95
112
5^8
19-3
9^7
249
■115
2-72
■42
5^20
■81
2
11-3
59
19^5
9-8
25^0
■12
2-78
■43
5^26
■82
7^26
21
ir7
6
19-6
99
25-2
■125
2-84
■44
5^32
■83
7^31
2-2
U-9
61
19^8
10
25-4
1
76
HYDRAULICS
Table II. — Imperfect and Partial Contraction for Large
Ebotangular Orifices in Thin Walls. (Art. 3.)
p
Values of ^.
Kemarks.
3
2-67
2
1
•5
Approximate Values of -.
If-T is not the same at all
d
parts of the border of the
■25
1
1-000
1-002
1-006
1-015
1-04
•50
1
1-001
1-003
1-013
1-030
1-13
orifice its mean value is to
be taken. The figures for
•75
1
1-001
1-004
1-019
1-045
1-28
— = 1 and -5 are only approxi-
1
1
1-002
1-006
1-025
1-060
1-56
mations. As —approaches
zero Ci increases rapidly.
Table III. — Co-efficients of Correction for Velocity
OP Approach. (Art. 5.)
(c„=-97. m=l-0.)
(1)
(2)
(3)
(4)
(B)
(6)
A
a'
a'2
vl2
l-c....-
1
V'-i
^1-.%-;
or Co.
1-33
-5625
-529
■471
■6ST
1456
1-5
•4444
•418
-5S2
•7g;5
1-311
2
-2500
•2.'?5
-765
■875
1143
2-5
-1596
•150
-850
■922
ro72
3
-nil
•104
-896
•!)47
r056
5
-0400
•038
•962
■981
1019
10
-0100
•010
•990
•995
1005
15
-0044
-004
•996
■997
1003
20
•0025
-0024
•9976
•999
1001
ORIFICES
77
Table IV. — Co-efficients of Discharge for Circular
Orifices in Thin Walls. (Art. 8.)
Head.
Diameter of Orifice in Inches.
■25
■60
■75
1
1-6
2
2-S
6
9
12
Feet.
•17
•25
■5
•75
1
l-i2
1-5
1-83
2
2-5
3^75
•683
•680
•669
•660
■653
•645
•643
•638
•637
•635
•629
•663
•657
■643
•637
•636
■624
•623
•621
■646
•632
•623
•618
■614
•613
•640
■626
■619
•612
•608
•618
■612
■606
■603
•612
•606
■601
■699
■610
■604
•600
■598
•597
■694
•592
For a square orifice add •005 to the above figures for same dia-
meter and head.
The first five lines of the table on page 56 show that c for an
orifice ■t feet square averaged about 'Gli under low heads. This
value is consistent with the above figures. It was increased by
perhaps '024 because of incomplete contraction, but it may have
been decreased owing to the submergence of the orifice.
Table V. — Co-efficients of Discharge ^ for Sharp-edged
Ee-entrant Tubes. (Art. 13.)
Diameter (Inches)
Co-efficient
•125
•91
■250
•87
■375
•85
■50
•83
•75
•81
I'O
■79
1^5
■77
20
•76
2-5
•75
The length of tube was in each case 2-5 diameters. The heads
were -5 ft. and upwards. The co-efficient showed no tendency to
vary with the head.
As in the case of orifices in thin walls, c tends to become 1 '0 for
an indefinitely small orifice.
■■ Bilton's co-effioients (i/m. Froc. Insl. C.E., vol. clxxiv.).
78
HYDRAULICS
Table VI. — Co-efficients of Discharge for Rectangular
Orifices, One Foot wide, in Thin Walls. (Art. 8.)
(1)
(2)
(3)
(4)
(5)
(6)
(V)
(8)
(9)
Height of Orifice in Feet.
■126
•25
•60
"75
1
1-5
2
4
Feet.
•2
•634
•3
•4
•634
•632
•632
•621
•633
•5
•633
•632
•619
•615
■6
■8
•633
■633
•632
■619
■618
•613
•612
•610
•606
■630
•632
1
■632
•632
•618
•612
•605
•624
1-25
•631
•632
•618
•611
•604
•624
■632
1-5
•630
■631
•618
•611
•604
■619
•627
2
2-5
3
•629
•628
•627
•630
•628
•627
•617
•616
•613
•610
•605
•605
•617
•615
•613
■628
•627
•619
•645
•637
•610
•610
■605
4
•624
•624
•614
•609
•605
•611
•616
•630
6
8
•615
•609
•615
•607
•609
•603
•604
•602
•602
•601
•606
•602
•610
•618
•610
•604
10
20
•606
•607
•603
■604
•601
•602
•601
•601
•601
•601
•601
■601
■602
•602
■604
•605
30
•609
•604
•603
•602
•601
•602
•603
•605
40
•611
•606
•604
■603
•602
■603
•605
•607
50
•614
•607
•605
•604
•602
•603
•606
•609
Table VIL— Co-efficients of Discharge for Small
Orifices (area -196 square inch) in Thin Walls. (Art. 8.)
Head.
Foot.
1
2
4
6
10
14
20
Equi-
lateral
triangle,
base
upward.
Square ♦
with sides
vertical.
Circular.
Rectangle with
long side
horizontal.
Remarks.
4tol t
16 to 1 t
•636
•628
•623
•620
•618
■618
•616
•627
•620
•616
•614
•612
•610
■609
•620
•613
•608
•607
•605
•604
•603
•643
•636
•629
•627
•624
•622
•621
•664
•651
•642
•637
•633
•630
•629
* With diagonal verti-
cal c is about •OO 14 greater.
t With long Bide verti-
cal e is abovit ^0014 less.
X With long side verti-
cal c is about 0005 less
for heads up to 10 feet,
and about ^0005 more for
the greater heads.
ORIFICES
79
Table VIII. — Co-efficients of Discharge for Submerged
Orifices in Thin Walls. (Art. 10.)
Head.
Size of Orifice in Feet.
Circle
•05 ft.
Square
•06 ft.
Circle
•1 ft.
Square
•1ft.
Rectangle
•05 ft. X -3 ft.
Feet.
•5
■616
■620
•602
■609
■622
1
■610
•615
■602
■606
■622
1-5
•607
■612
•601
■605
■621
2
■604
■609
■600
■604
■620
2-5
•603
■608
■599
■604
■619
3
■602
■607
■599
•604
■618
4
•601
■607
■599
•605
Table IX. — Co-efficients of Discharge for Cylindrical
Tubes. (Art. 12.)
Head.
Diameter of Tube in Inches.
-25
■50
1
s
Feet.
•5
2
22
•84
■83
■83
■82
•82
■81
■80
■80
■80
80
HYDRAULICS
Table X. — Co-efficients of Correction
FOR Vertical Orifices with Small Heads. (Art. 19).
(1)
(2)
(3)
(1)
(6)
(6)
(7)
(8)
(9)
Head
over
centre of
square
orifice
with
sharp
edges.
Head over
centre of
bell-
mouthed
orifice or
of vena
contracta
for sharp-
edged
orifice.
Rect-
angle.
Circle or
semicircle
with
diameter
vertical.
Tri-
an^le
with
base
up.
ward.
Tri-
angle
with
base
down-
ward.
Semi-
circle
with
dia-
meter
up-
ward.
Semi-
circle
with
dia-
meter
down-
ward.
Remarks.
52i)
•64£>
■781)
•92i)
144Z>
■50Z>
•52D
•557)
■60Z>
■70Z>
•80Z>
■9QD
lOi)
1-2D
r5X)
20i)
2oXl
3'0Z>
4 01)
•943
•950
•957
■966
•976
■982
•986
■989
•992
•995
•997
•998
•999
•999
•960
•965
■970
•975
•982
•987
•990
•992
•994
•997
■998
•999
■999
1000
•924
•996
•979
•998
■937
•996
•965
•997
The co-
efficients have
not been
worked out in
detail for tri-
angles and
semicircles,
but can be
easily esti-
mated from
the figures
given in the
first and
tenth lines.
"When the
head is
greater than
D the co-
efficients for
orifices of all
shapes are
nearly equal.
CHAPTEK IV
WEIES
[For preliminary information see chapter ii. articles 4, G, 7, 14, and 15]
Section I. — Weirs in General
1. General Information. — The following statement shows a few
typical kinds of weirs, and gives some idea as regards the co-
efficients. Further co-efficients will be given in subsequent
articles, and from them the values for many cases occurring in
practice can be inferred, but the varieties of cross-section are
innumerable, the co-efficients vary greatly, and generally can only
be found accurately by actual observation. When the length, /, of
a weir is great relatively to H, it makes little difference whether
there are end contractions or not.
To ensure complete contraction iron iiled sharp should be used
for the upstream edges with small heads. For heads of over a
foot planks or masonry may be used.
Since the inclusive co-efficient C increases with H, it follows that
when there is velocity of approach Q increases faster than 11^. If Ii
is doubled Q is about trebled. To double the discharge //must be
multiplied by 1 '5. If a given volume of water passes in succession
over two similar weirs, one of which is three times as long as the
other, the head on it will be half that on the other. If a volume of
water, passing in succession over two weirs, alters, the heads on both
will alter in nearly the same ratio. These rules are only approxi-
mate, and when there is no velocity of approach they are somewhat
modified. To facilitate calculations the values of H^ correspond-
ing to different values of H are given in table xi.
Smith states that with low heads such as '2 foot the discharge
may be affected by a change in the temperature of the water of
30° Fahr. If the water is disturbed by waves or eddies the
discharge is probably reduced, unless ' baffles ' are used ^ to calm it.
In the sheet of water passing the edge of a weir in a thin wall
' Or grids. They should not be so near to a weir or orifice iis to interfere
with the flow of approach.
F
82
HYDRAULICS
Vakious Kinds of Weirs and their Co-efficients.
Type of Weir.
Fio. 6.5.
Thin Wall.
Fia. 66.
Flat toij, vertical face and back.
Fig. 67.
Steej) back and sloping face.
Fig. 68.
Steep face and sloping back.
Dimensions of Weirs for
which Co-efficients are
quoted.
Feet.
1-64
as
Feet.
2-46
Manner in
which Co-
efficient
varies as
Head
increases.
1-31 verti
cal
1-64
t.-rr
W<5S#^^is&li^5"SN?:SSS V S^
Pig. 89.
Eounded.
1-64
•33
to 1
•33
1^64
■67
Increases
slowly
verti- ^54 Increases
cal rapidly
verti- •To Increases
cal
verti-
cal
5 to 1
"01 Increases
85
Increases
These woira are some of the types used by Baziu in his experiments
1 hero wore no end contractions. The co-efficient O includes the allowauoe
tor velocity of approach.
WEIRS
83
the velocity is greatest at the lower side, but with a broad-topped
weir the friction on the top reduces the velocities nearest the weir.
In every case the initial horizontal velocity of the whole sheet may
be taken to be § J'igH, and the path of the sheet calculated as for
orifices (chap. iii. art. 7). Fig. 70 shows a separating weir as used for
water-supplies of towns.
After heavy rain the
water is discoloured
and H is great, so that
the sheet falls as shown
and the water is con-
veyed to a waste chan-
nel. At other times
the water falls into the
opening K and is con-
veyed to the service
reservoirs. The velocity
at the ends of a weir is
generally less than else-
where, and it increases
up to a point distant about ZHivam. the ends. The pressure in the
water passing over the crest of a weir is less than that due to the head.
The following statement shows the chief experiments on weirs
in thin walls : —
Fig. 70.
No. of
Head.
Distance
Observer.
Obser-
vations
made.
Length
of Weir.
Height
of Weir.
state of
Contrac-
tion.
of Measur-
ing Section
from Crest
of Weir.
From
To
Feet.
Feet.
Feet.
Feet.
Feet.
Francis,
46
10
•6
1^6
4-6
Com-
6^0
„
19
10
•6
1^0
2
plete
60
»>
6
4
•7
10
4^6 & 2^0
or
6^0
Smith,
12
2-6
•6
1-7
3^8
nearly
7-6
Lesbros, .
21
1-77
•1
•6
1^8
com-
11^5
Poncelet&Lesbros,
6
■66
■08
•7
1-8
plete.
11-5
Fteley & Stearns,
54
2-3 to 5
•15
•94
3^6
1 Vari-
j able.
60
Lesbros, .
34
•66
■06
•7
\-s
115
Francis, .
17
10
■7
VO
4-6
6
Fteley & Stearns,
10
19
•5
16
66
End
6-0
J,
30
5
■07
■8
3-2
con-
6^0
Lesbros, .
14
■66
■06
•8
1-8
■ trac-
\\-5
Baziu,
295
656
■23
1-0
3^7 to -8
tions
16^4
38
3-28
■23
r3
3-3
absent.
164
j»
48
1-64
•23
1^8
33
16^4
84
HYDRAULICS
Fig. 70a.
2. Formulae. — The ordinary weir formula (equation 1 1, p. 15)
and the other formulae deduced from it are defective in form. It
has been said that
the head ND
(Fig. 71) ought
to be taken into
account, the dis-
charge of the
weir being con-
sidered to be that
of an orifice
whose bottom
edge is C and top
edge D. But a
weir is not an
orifice. The sur-
face contraction
makea the cases
different. It is possible that the head H should be measured from
F and not from G, and it is unlikely that ^H really represents
exactly the head corresponding to the mean velocity. The case is
really one of variable flow in a short channel, and it would probably
be treated as such if it were practicable to observe the heads at D
and F. As it is, shortcomings in the formula are made good by
the values given to the co-efEcieuts.
In all weir formula: m can be written for \c, and this plan is
adopted by Bazin ; but c is the true co-efficient expressing the
relation between the actual and the theoretical discharge, and
it is desirable that c should be used both in formulae and in
tables. Since 5 ^2;/ = 5 35 this figiu-c can be used in calculations
instead of 8 02, and multipliciition by f is thus unnecessary. The
values of \c J'2g corresponding to different values of c are given in
table xii. and denoted by K. They are the discharges per foot run
over a weir with II=\ foot. Engineers frequently condense the
formula by using K instead of c, but the value of c should not be
lost sight of.
Fig. 71.
WEIRS 85
3. Incomplete Contraction. — From a comparison of the co-
efficients obtained for various weirs in thin walls, Smith arrives
at the formula
«,= c(l + -16|)
where c^ and c are the co-efficients for two equal weirs, one with
partial and one with full contraction. P is the complete perimeter
of the weir, that is l+'2,H, S the length of the perimeter over
which the contraction is suppressed. This formula applies for
heads ranging from -3 foot to 1 -0 foot ; it is not exact, but may be
used for finding co-efficients not otherwise known.
When the contraction is imperfect,^ whether or not the margin
is sufficient to give a negligible velocity of approach, the formula
arrived at by Smith is
0+4)
where c^ is the co-efficient for the weir with imperfect contraction,
S' the length of its perimeter on which the contraction is imperfect,
and X is as follows, d being the least dimension of the weir and G
the width of the clear margin.
%= 3 2-67 2 1-50
a
x= -0016 -005 -025 -06 -16
When the contraction is imperfect over the whole perimeter S'=^P,
and when
^= 3 2-67 2 1 -5
a
the increase in c per cent.
= -16 -50 2-5 6 16
But when ;S^ is a very large fraction of P, or when S'—P and -;
is very small — that is, when there is not much contraction left
except at the surface — the rules become of doubtful application.^
4. Flow of Approach. — Bazin observed some surface-curves for
weirs 3-72 feet and ri5 feet high, and for each weir with several
heads ranging from '5 feet to 15 feet. He finds «/ (Fig. Vl)'''to be
in every case about 3H, but the upper portions of the curves are
so flat, especially for the lower heads, that it is impossible to say
exactly where they begin. Observations made by Fteley and
Stearns, with H nearly constant and difierent values of G, give
results somewhat similar to Bazin's, but when G is less thanij, yis
' For definitions of ' partial ' and ' imperfect ' see chap. iii. art. 3.
^ liounding of crest and sides may increase c some 20 per cent. "When con-
traction is thus suppressed the surface contraction doubtless increases, of. chap,
iii. art. 3. ° AN=y.
86 HYDRAULICS
about 2-5G. The above indicates the proper distance from the
weir to the measuring section. In weirs with end contractions G',
the distance of the end of the weir from the side of the channel
must be used instead of G if it exceeds G. In a weir with a long
sloping face Smith found y to be 40 feet with H=7-24: feet.
The fall ND or F for weirs in thin walls is generally between
H H •
jx and — It is much greater with broad-topped weirs. In the
above experiments with weirs in thin walls -^ was found to be as
follows : —
G=3-5Q 1-7 -5 3-72 1-15 feet.
.0^= -614 -606 -564 -5 to 1-5 -5 to 1-5 „
^= -148 '145 -114 -149 -143
n.
Fteley and Stearns. Bazin.
Some other values are
H= '68 -37 -20 -08 feet. ] Poncelet and Lesbroa, weirs
:^_.0S •11 -15 -25 f in thin walls, full contrao-
ff~ } tion, length -66 foot.
And for flat-topped weirs
11= -5 -1 -5 -1 -5 -1 feet.
^=■21 -28 -29 -40 -64 -67
Top width : "5 inch. 2 inches. 3 inches.
According to Smith F is somewhat greater in weirs with no end contractions
than in others, and increases slightly with I.
Fteley and Stearns found that just upstream of a weir the pressure, at least
near the bottom, is greater than at the same level further upstream. Gener-
ally the difl'erence is nearly as h or -— , and it also increases as O decreases.
It never exceeded the amount due to a head of "03 foot, and was generally
much less.
5. Velocity of Approach. — The ordinary formulse for weirs with
velocity of approach are
= mlJ%j{H+vJi,)i} ' ' ' ^ ''
By using a variable co-officient of correction c„ we obtain the inclu-
sive co-efficients C=rc„ and M—mc,,.
The formuhu with inclusive co-efl5cients are
Q=iClj2gm\
WEIRS 8/
For weirs in thin walls with complete contraction equation 42 is
not ordinarily suitable, because while the values of c are known
and tabulated those of G are not known, and if calculated for many-
different values of v would fill a formidable set of tables. But for
other kinds of weirs G is often known as well as or better than c. In
these cases, and also in cases where Q is to be measured for some
particular weir, and the co-efficients ascertained and recorded,
equation 42 is eminently suitable.^
Where c is not known the use of c„ renders the adoption of the
indirect or tentative solution unnecessary in certain cases, and so
saves trouble (see examples 1 and 5). It is not convenient to give
a formula, as in the case of orifices (equation 22, p. 48), for
calculating c„, because equation 1 1 gives Q and not v. In order to
find V it would be necessary to separate c into Cj, and c«, and these
quantities are not properly known. Values of c„ have, however,
been found by working out various cases, and are given in table
xiii. for two values of c. Others can be interpolated if required.
The excess of c^ above 1 '0 is nearly as c^, and for a given value
of c nearly as n. The co-efficient c„ may be used either for solving
ordinary problems or for obtaining values of G from cot M from m.
The inverse process of finding c from C or m from M is as
follows : —
Since Q=vA,
Therefore from equation 42 JL = '"^f ^^'~J^ • ' " (*^'^^-
But Q=mlJTg(H+n^y
Since the last term in the brackets is small compared to the first
term, the expression in brackets is nearly equal to 1+ %.— ^_
Adopting this value and substituting from equation 42a
G=mZV2^a'*(l + 2»^''2i") • • • (43).
From equations 42 and 43
M
1 + 1.^4
(44).
It is of course impossible to observe either m or n directly. The
' Provided the bed is not liable to alter, see example 5.
88 HYDRAULICS
observations give M directly, and either m or n can be found by-
assuming a value for the other. Generally m is assumed or
deduced from its values for a similar weir with no velocity of
approach, and n is then calculated. When the length of a weir is
the same as the width of the channel of approach and G is the
height of the weir : equation 44 becomes
m=--- -^^ . . . (45),
and in this form is given by Bazin.
On the assumption that the effect of the energy due to the velocity of
approach is the same as that of raising the water-level by a height AK
(Fig. 71) equal to -,j— , the discharge is the same as that through an orifice
with heads KA and KE, and the old form of equation was
which is similar to equation 35, p. 70. This equation cannot be of the true
theoretical form, chiefly because the original weir formula (equation 11,
p. 15) is not so. It would, however, be right to use it, as the best attempt
at a theoretical formula, if there were any advantage in doing so. But the
last terra h^- is generally small and often minute, while the formula is more
complicated than equation 12. The method of allowance for " is largely
empirical, and it is better to use the more simple formula 12. With this
formula n might be expected to be somewhat less than unity.
From article 7, chapter ii. it is clear that for weirs with velocity
of approach the contraction may be either perfect or imperfect.
When it is imperfect the increase of discharge is due partly to the
energy of the water, represented by -- — and partly to reduced
contraction due to smallness of the margin. The value of n from
both causes combined has been found to be, for weirs in thin walls,
from I'O to 2'5. Smith rightly separates the two causes, and,
discussing various experiments, concludes that n should be 1 "4 for
weirs with full contraction, and 1 'SS for weirs with no end con-
tractions. The eflfect of reduced contraction, if any, was estimated
separately, but the allowance made in the cases of weirs with no
end contractions was not quite sufficient according to the rules
given in article 3 above, so that n was a little overestimated, and
Smith himself suggests that this may be so. Since Smith wrote,
the results of Bazin's experiments on weirs with no end contrac-
tions have appeared. Owing to their general regularity and
extent thoy are entitled to great ^veight. By analysing them on
WEIRS
89
Smith's principle it is found that n varies from -86 to 1'37, and
averages about I'l. For moderate velocities of approach Q depends
only a little on n (see table xiii.), and it is not worth while to give
here the detailed analysis. ^ Bazin himself gives I'Si as the mean
value of n, but this includes the effect of reduced contraction.
Both sets of experiments, namely Bazin's and those discussed by
J
Smith, include high velocities of approach, the ratio — being
ct
sometimes only 1"6. For weirs with full contraction the experi-
ments discussed by Smith are not numerous, and his resulting
figure 1-4 somewhat doubtful. It seems high in comparison with
the others, and may be put at 1 '33.
The variations in n, and especially its exceeding the value 1"0, are not
easy to explain. A weir is usually in the centre of a, channel, and the
average deflection of the various portions of the approaching stream is
then a minimum, especially if its greatest velocity is also in the centre, so
that a large proportion of the water flows straight. In a weir so placed ?i
will be a maximum ; but this is no reason for its being greater than unity.
The whole of the water, and not only the quickest water, has to pass over
the weir. At the approach section the velocity distribution (chap. ii.
art. 21) is normal. The total energies of the various portions of the
stream may (chap. ii. art. 10) exceed the energy due to v, but the differ-
ence is probably only a few per cent., and nothing like 33 or even 20.
Moreover, some little energy must be lost in eddies between the approach
section and the weir. Thus in no case will the available energy appreciably
exceed that due to -^. A high velocity of approach does not of itself
reduce contraction. The high velocity occurs in the portion EB (Fig. 71)
as well as in A E. With an orifice in the side of a reservoir a high velocity
does not cause reduced contraction, but rather the contrary. The surface
curves for weirs do not indicate any reduced surface contraction when v is
high. Reduction of the clear margin is allowed for separately ; and there
are high values of n for cases in which the clear margin is ample.
It is probable that the deviations of n from unity are chiefly
due to the incorrect form of the equation used. If a curved crest
FC is added, the flow will not be appreciably affected, but the
head will now be H' instead of H. The co-efficients of the two
weirs must be such that cH^=c!H''^. Suppose A now reduced
so that V becomes considerable, then c(Ii-\-nhY must equal
TTf
c'(n'+n'h)i, and this occurs when n'=n-^. If c is -60 and c' is
H n
•80 (values likely to occur in practice), -. = _ ^ = 1 -2. Thus it
^ It will be found in Appendix C.
90
HYDRAULICS
can be seen how imperfections in the formula may cause n to
change, and also that for a weir with a sharp edge n is greater
than for a rounded weir.
The following values for n seem suitable for weirs situated in
the centre of the stream : —
Weirs with end
contractions.
Weirs withont end
contractions.
Weir with sharp edge,
Rounded weir, .
1-33
11
)-2
10
For other kinds of weirs the value can be estimated. For a weir
not in the centre a reduction can be made. When the edges are
sharp, and the margin insufficient for complete contraction, an
additional allowance for this must be made by the rules of
article 3.
Section II. — Weirs in Thin Walls
6. Co-efB.cients of Discharge. — The chief experiments on weirs in
thin walls, except Bazin's, have been analysed by Smith, who has
prepared tables of the values of c at which he arrives, and his
results somewhat condensed are shown in tables xiv. to xvi., but he
notes that when H is less than -2 foot the figures are not reliable.
Those cases which are marked (?) Smith considered doubtful,
owing to the absence of observations for such cases. For the
others he gives the probable error as only -3 per cent. It is of
course known that end contractions reduce the discharge, and that
their effect increases with H and decreases with I. Smith in his
analysis considers all the experiments (except Bazin's) mentioned
in article 1 — those with and those without end contractions and
those having various degrees of contraction — together, and to a
certain extent infers one set of values from the other.
But further observations have been made by Stewart and Long-
well (Trans. Am. Soc. C.E., vol. Ixxvi.) on short weirs with full
or nearly full contraction. The weirs were only one foot high, and
for this reason the figures, for the cases where H was highest, have
been slightly reduced by Gourley and Crimp (Min. Froc. Inst.
C.E., vol. oc). Their figures — in some cases again slightly altered
so as to accord with the rules of art. 3 — for weirs less than 3 feet
long are shown in table xiv., and supersede Smith's figures, some of
WEIES 91
which he himself considered doubtful. For the 3-foot weir their figure
for a 2-foot head is shown ; for smaller heads their figures exceed
Smith's by about "004. The co-efficients in table xv., obtained from
experiments by Castel, do not, for the smaller heads, accord with
those of table xiv. and are probably incorrect. Such very short
weirs are not important, measurements by orifices being better.
For weirs with no end contractions Bazin obtains figures differing
from those of Smith. Smith's co-efficients attain a minimum
as R increases and then increase, but Bazin's decrease as long
as K increases. Smith's co-efficients increase as I decreases,
but Bazin's are constant. The discrepancies are important because
of the different laws which they indicate, and because of the
high standard of accuracy obtainable with weirs in thin walls.
The methods used for observing the head are described in chapter,
viii. article 6. Bazin's measuring section was (art. 1) 16 '4 feet up-
stream of the weir. It has been suggested that the surface fall in
thia length caused an error. Calculations show that the error
must have been inappreciable. Whether Bazin's weir had a length
of 6'56 feet, 3'28 feet, or 1 -64 feet, his values of c come out the same.
Bazin considers that in Fteley and Stearns' experiments baffles were
placed too near the weirs. Bazin's co-efficients are confirmed by
experiments made by Kafter ^ and to some extent by experiments
made at Wisconsin University.^ They should be used for weirs
1 '5 to 8 or 9 feet long, without end contractions. For longer
weirs Smith's figures should be used.
The detailed values of Bazin's co-efifioients given in table xvi. are,
owing to Bazin's values of n not being accepted (art. 5), slightly
higher for the greater heads than the values arrived at by Bazin
himself. They accordingly differ less from Smith's figures. Bazin
calculated c, or rather m, for heads ranging from -16 to 1 '97 feet,
but his actual observations were within the range shown in table
xvi. Bazin also gives a complete table of the values of M, and from
it table xviii. giving values of C has been framed.
It has been found that when there are no end contractions the
sheet of water after passing the crest of a weir tends to expand
laterally, except when B. is less than -20 feet, and the side-walls
have usually been prolonged downstream of the crest, openings
for free access of air beneath the sheet being left. If the sides
are not so prolonged c will be increased about -25 per cent, when
H — —', and more or less as // is more or less. It also appears
that in such weirs moderate roughness of the sides of the channel
has no appreciable effect on the discharge.
' Hydraulic Flow Meviewed (Barnes), Table 8.
92 HYDRAULICS
Eegarding triangular weirs in thin walls, observatiors have been
made by Gourley and Crimp (pp. cit.). They adopted a formula
involving H^'", but their figures enable c in the ordinary formula
(equation 54, p. Ill) to be calculated. The figures are given on
p. 96. They confirm previous figures obtained by Thomson, by
Barr {Engineering, vol. Ixxxix. p. 473), and by Gaskell (Min. Proe.
Inst. C.E., vol. cxcvii.), and they are independent of the side slopes
of the weir which varied from |^ to 1 to 1 to 1.
7. Laws of Variation of Co-efficients. — The following laws,
governing the variation of the co-efiScient for complete contrac-
tion, are apparent : —
(1) For cross sections of similar shapes, i.e. a given ratio of I
to H, c is less as the section is greater.
(2) In the short weirs the section is sometimes square, i.e. l = H
nearly. In these cases c tends to increase or become constant when
H exceeds I.
(3) For the other rectangular weirs c decreases as S increases.
(4) For a triangular weir c is somewhat less than for a rectan-
gular weir with the same values of I and H. The contraction in
the acute angles is hindered (chap. iii. art. 8), but the surface con-
traction is probably increased because the surface stream has only
narrower streams to hold it up.
Some of the laws are similar to those for orifices in thin walls,
but the surface contraction in weirs creates a great distinction
between the two cases.
8. Flow when Air is excluded. — With four weirs in thin walls,
of heights 2'46 feet, 1'64 feet, 1'15 feet, and -79 foot, further
observations were made by Bazin, the access of air beneath the
falling sheet being prevented by the closure of the openings which
had been left for that purpose. The following statement shows
the results noticed. The pressures under the sheets were ob-
served, and the discharge was found to increase as the pressure
decreased.
An interesting point for consideration is the conditions under
which the different forms are assumed. This is stated by Bazin,
and is shown in the above statement. AVith weirs not exactly
similar to those of Bazin, it may be difficult to say when the
various changes will occur, but it will at least be possible to
foresee thorn and to take some account of them when they do
occur. The occurrence of the form called 'drowned underneath'
will obviously be affected by the condition of flow in the down-
stream reach. One lesson to be learnt is, that if complications
are to be avoided and discharges accurately inferred the free
access of air under the sheet is essential.
WEIES
93
9. Remarks. — When discharges are to be measured by weirs
those without end contractions may be easier to construct. For
measuring the very variable discharge from a catchment area, a
weir in a thin wall has been used with a central notch (Fig. 71a)
which can deal with small discharges and so avoid very small
heads (Min. Proc. Inst. G.E., vol. cxciv.). It would seem best to
construct ah de so as to have no contractions there. Otherwise
when the water covers the whole crest, as in the iigure, the central
portion of the water is subject to contraction on ab de, but not
on be ef, and the co-efEoient of the central portion must be doubtful.
A triangular weir would probably be best if c were determined
for large triangles.
Cippoletti's formula is § = 3*367 I Hi (c=-63) and the weir is
trapezoidal, the side slopes being |- to 1 and I being the bottom
width. The sloping ends counteract the increasing effects of the
end contractions as H increases, and c remains constant as long
as His not > -. It is not known that the formula is accurate when
o
Z>9 feet or H>\-^ feet. When Z is 3 feet or less, the formula
is known to be accurate to within, say, 2 per cent.„within the above
limits, and to be inaccurate outside them. For instance, it gives
results about 30 per cent, too small when l = \ foot and H=2 feet
(op. cit., vol. cxciii.).
Regarding trapezoidal weirs in general (Fig. 71b), let q^ be the
discharge of abn and q^ that of dbef when they are separate and each
has full contraction. Gourley and Crimp found (op. cit.) that, for
abeg, Q^q^ + q^ *'° within 1 or 2 per cent. The length be varied
from "25 foot to 3 feet, the side slopes from i to 1 to 1 to 1,
and H from '2 foot to 1 foot. For cbeg alone the discharge is
probably ^j- When it is joined to abc contraction on be ceases, but
three acute angles — at b and c — are abolished. Thus for small
trapezoidal weirs in thin walls with full contraction the special
formula (art. 16) is not needed.
Expeslments by Flinn and Dyer (Trans. Am. Soc. G.E., vol. xxxii.)
on trapezoidal weirs with lengths up to 9 feet and H up to 1-4
feet — side slopes;^ to 1 — show some fluctuations and are not accurate
enough to test the above law further.
w////////////^
Fig. 71b.
Fig. 71a.
94
HYDKAULICS
Reference
to Fig.
Fig. 72.
Name given
to Case by
Bazin.
Adherent
sheet.
Description of
Case.
Fig. 73.
Fig. 74.
Fig. 75.
Depressed
sheet.
Sheet
drowned
under-
neath.
Sheet in con-
tact with weir
and no air
under it, or it
may spring
clear from the
iron plate, en-
close a small
volume of air,
and then ad-
here to the
plank, or it
may adhere to
the top and
bevelled edge
and then spring
clear, enclosing
air as in the
case following.
Conditions
under which
it occurs.
Under small
heads.
Air partly
exhausted by
the water and at
less than atmo-
spheric pres-
sure. Water
under sheet
rises to higher
level than that
of tail water.
Water under
sheet rises to
level of crest
and all air is
expelled.
(a) Wave at a
distance.
(b) Wavrcov-
eringfoot
of sheet.
When case 1
does not occur,
or when it oc-
curs and jff is
increased. The
change occurs
abruptly.
When H is
further in-
creased so that
H is not less
than about '4 G.
When the
fall H+H^ is
greater than
about J 6.
When the
fall H+H^ is
not greater
than about J (V.
For a given
head H the
greatest value
of H« is ?(?-
EfTect on tlie Co-efficient
of Discharge, C.
C may possibly exceed
that for a free sheet by
33 per cent.
C is higher than for
free sheet, generally only
slightly, but it may be
10 per cent, higher when
it is on the point of as-
suming form ' drowned
underneath.'
Value
of-g--
•05 .
■40 .
■50 .
■60 .
•80 .
1^00 .
1-20 .
1-10 .
1-60 ,
Value
C
of-
l-2'2 -^
1^19
1-13
. 1-m
. 104
1-005
. -98
. -96
. -95 ;
O is the
co-efficient
for a free
sheet and
C for the
case in
question.
The level of the tail
water alfects the discharge,
and approximately
g:=(l^05 + -15§5)...(46).
See also article 13.
WEIRS
95
Fig. 74.
Fig. 75.
96
HYDRAULICS
Francis found that end contraction might be allowed for by
considering the length of the weir to be reduced by -2011, that is,
by substituting (1—-2E) for I in equation 11, page 15. He
found that with the formula thus modified, the co-efficient, pro-
vided I is not less than 3H or iE, is nearly constant, its value
being -620 to -624, and averaging -623 for heads ranging from 5
to 19 inches. Results obtained by this formula are liable to
differ by 1 or 2 pgr cent, from those of the ordinary formula with
the co-efficients of table liv. It is not known that the formula
is correct when l> 10 feet. When c = -623, Q = 333 I III
In either formula — Cippoletti or Francis —velocity of approach
can be allowed for (equation 12, p. 15). Both formulae are useful
attempts at simplification while adhering to simple indices. Further
experiments may enable a Cippoletti weir to be designed with the
sides curved, the slope altering as H increases so that c remains
constant.
CO-EFFICIBNTS FOR TRIANGULAR WeIRS IN THIN WaLLS. (Art. 6.)
s=-\.
■2
■3
■4
•5
•6
■7
•8
•9
1 foot
c=-616
•605
■597
•691
•587
■584
■581
•579
•677
•575
Section III. — Other Weirs
10. Weirs with, flat top and vertical face and back. — Generally
the water at B (Fig. 76) holds back that upstream of it, and the
discharge is less than for a weir in a thin wall under the same
head. It is a sort of drowned
weir, B being the tail-water
level.* At A there is eddying
water. When // is about
1-6PF to 2/r— /F being the
top width — the sheet springs
clear from the top, and the
case becomes a weir in a thin
wall. But if the sheet nearly
touches at 6^ (Fig. 77) the water gradually abstracts the air, and
the sheet is pressed down, touches at C, and Q is slightly greater
than for a weir in a thin wall. Table xvii. (prepared by Fteley and
Stearns) shows the corrections to be applied to c, the co-efficient
for weirs in thin walls, in order to give c„ the co-officient for weirs
with flat top and vortical face and back. The corrections apply
• Soe also art. 15.
Fia 76.
Fio. 77.
WEIRS 97
strictly only to weirs ^¥ithout end contractions, but may be used
for others.
Bazin made numerous observations on weirs of this kind, and
his results are shown in table xix. Some observations made at
Cornell University, in the United States of America, are included.
Some of them contained sources of error. The method of observing
the head (chap. viii. art. 6) admittedly caused error. Some of the
figures were corrected after further observations and calculations
( Weir Experiments, Co-efficients and Formula;, E. E. Horton). As
to those not so corrected, it was concluded that they were correct to
within 6 per cent. They are marked (?) and are given as approxi-
mations and because other co-efficients for some of the heads are not
available. These remarks refer also to tables xx., xxi., and xxii. The
Cornell weirs are those 4 feet to 5-3 feet high. Eesults of experiments
by the Geological Survey, U.S.A., on fiat-topped weirs 11'25 feet
high are also included. The various figures are consistent.
Bazin gives the following formula for obtaining C^, the inclusive
co-efBcient for such weirs, from C, the co-efficient for a weir in
a thin wall.
|»=-70 + -185 J . . . (48).
The results given by this formula agree with the observed results
generally within about 2 per cent., but for the widths of 6'56 feet,
2-62 feet, and 1'31 feet the error may be 3 or 4 per cent. They
also agree with Fteley and Stearns' results within 1 or 2 per cent.
When if was increased to about 2/Fthe sheet sprang clear, but
if H was gradually lowered the sheet remained clear till E was
about l-6?F. Between these limits it was unstable. When the
sheet springs clear the above formula of course is not needed. The
thick lines in the table mark off the cases when S" was less than
SW C
2W. While if varies from -— to 'iW, the ratio _ii!.may change
from -98 to 1 -07 if the sheet remains attached to the crest.
When air was excluded depressed and drowned sheets occurred
under somewhat similar conditions to those with weirs in thin
walls. Eemarks regarding them are given in table xix. Their
occurrence sometimes preceded and sometimes succeeded that of
detachment of the sheet from the back or top of the weir, and
rendered the conditions very complicated.
11. Weirs with, sloping face or back. — Bazin's chief results for
weirs of this class are given in tables xxi. and xxii., and the
as
HYDRAULICS
Cornell results are included. Table xxi. contains the cases where
the back of the weir was steep, so that the sheet generally sprang
clear of it. Apparently no air openings were left, and the adherent
depressed and drowned sheets often occurred. Table xxii. shows
the cases where the back slopes gradually. In these last the
stream flowing down the back is in uniform flow in an open
channel.' Weirs of this kind with back slopes about 10 to 1 are
used on some large canals in India and termed 'Rapids,' the
profile of the water-surface being as sketched in Fig. 68, page 82.
The flow at the crest is virtually that of a drowned weir. At the
foot there is a standing wave (chap. vii. art. 11).
In weirs of these classes there are several variable elements.
Pairs of cases in the tables can be compared in which only one
element varied, so that its effect can be traced. By studying these
cases and the tables generally it will be seen that C generally
increases as the height of the weir decreases, as the top width of
the weir decreases (but not so much for the greater heads), as the
upstream slope is flattened, and as the downstream slope is made
steeper.
12. Miscellaneous Weirs. — For a weir made of plank with a
rounded crest of radius R the discharge with head H is about the
same as for a weir in a thin wall with a head H'. The following
table is given by Smith ^ : —
H.
Values o{R.
25 in.
•60 in.
1 in.
Values of H'-Jf.
■116
•006
■004
003
•166
•014
•013
■015
•217
•021
■018
•28-1
•Oil
■029
■Oi'S
•351
•015
■02S
■039
•41
■014
■028
■044
•4 9
•015
■030
052
The chief results of the Bazin and Cornell observations on rounded
weirs are given in table xx.
1 But eoe art. 16. ° Hydraulics, ohap. v.
WEIRS
99
.:^Ia^
Fia. 78.
Fia. 783.
For a weir formed entirely by lateral contraction of the channel,
and having a crest length of 2 feet to 6 feet (Fig. 82, p. 107), c is
•65 to '73 and C is -70 to -78, being greater for the larger sizes.
l^or a fall (Fig. 79) in which there is neither a raised weir nor a lateral
contraction there is no local
reduction of the approach- H
ing stream due to eddies
or walls, and therefore no
local surface fall of the
kind ordinarily occurring.
The surface curve is due
to draw (chap. ii. art. 11).
If the slope AB\s not very-
steep the curve extends Fia. 79.
for a great distance.
If V is the velocity at DE near to BC, then V is both the velocity of
approach and the velocity in the weir formula, so that
If c= -79 and 0== -63 and m=l-0.
F'
=-Jc=23(//+»^)
V-=.
■28
1-
Igll
•28
^=•62 V2y^.
If the channel ^45 be supposed to be very smooth or steep the water-surface
HO will be parallel to the bed, but there will always be a short length OG
in which draw will occur. Tails of this kind occur at the ends of wooden
troughs and shoots. They were used on one of the older of the great
Indian canals, but the high velocity due to the draw caused such scour and
damage that raised weirs had to be added.
Section IV. — Submerged Weirs
13. Weirs in TMn Walls. — The following statement shows
the chief experiments which have been made.
observer.
Length
of Weir.
Upstream Head
Ih-
Downstream Head
Height of
Weir.
From
To
From
To
Francis,
Fteley and Stearns,
Bazin, . - .
Feet.
11
5
6^56
Feet.
1^0
•33
•19
Feet.
2-3
■81
1^49
Feet.
•24
•07
•79
Feet.
M
■80
1-2G
Feet.
5^8
3^2
■8 to 2-5
100
HYDRAULICS
The weirs were all without end contractions. The level of
the tail water was measured
at M (Fig. 80), which is theor-
jS3 =^^ etically wrong ;i the surging
" " of this water renders exact
measurements difficult. The
CO -efficients for submerged
weirs are not, in most cases,
well known, and exact results
cannot be expected from
them.
'^y////////J'///////////////////////////
Frn. 80.
Let 2', be the discharge through AB and jj through BO. Then
q.^z^clJ-lgH. H . . .
If c has the same value for both portions,
OT q = d J2gH (H, + ^) ■ ■
(49).
(50).
/ fi\
or q=d ^2^ [H,-^ J
(51).
(52).
The last two formulae are those for an orifice having a height
equal to the downstream head plus two-thirds of the fall. If
there is velocity of approach H+nh must be put for H and Si+nh
for H„ but //; is left unaltered.
Francis makes C2 = -92161, that is, he multiplies E^ in equation
52 by -921. Smith, discussing the experiments of Francis and
Fteley and Stearns, and reviewing a previous discussion by
Herschel, substitutes -915 for "921 and recommends the formula —
q=r;j2gjH+nh) (-915 H. + ^-iE+^'j .
(53).
This formula is for woirs in thin walls without end contractions :
c, is the co-efficient taken from table xvi. for the equivalent weir
with a free fall (that is, the weir with a free fall giving the same
discharge) and n is 1-33. The formula may be applied to weirs
with end contractions and the same co-efficients used if l—-2Hi bo
substituted for I.
If Q is the discharge for a free weir, and if H^ remains constant
while the tail water is raised by some cause operating in the
' See chap. ii. art. 6.
WEIRS 101
downstream reach, Q decreases very slowly till H^ is about ^.
The discharge through AB is the same as before, while the
velocity in BO is altered in the ratio /./ „ ■ The relative dis-
charges are as follows, c being constant and velocity of approach
being supposed to be negligible : —
-=•00
•25
•33
•50
•66
•75.
or^= -00
•3S
•50
l-Q
2^0
3-0;
a 52) =1-00
•974
•953
•88
'11
•69
I (equation 53) =rOO -945 -933 ^84 -71 -61.
Practically, this law is somewhat modified. Let it be supposed
that for the free weir there is ample access of air. As the tail
water rises above the crest the air is shut out. The under side
of the sheet springs up to a somewhat higher level than the crest,
but the surging of the tail water shuts out the air almost at once.
The sheet of water is pressed down, and the discharge instead of
decreasing increases a little. Practically it remains nearly con-
stant during a certain rise of the tail water and then decreases.
If the air passages become obstructed just before the tail water
rises to the crest level, Q will begin to increase then, but this does
not necessarily occur. Neither equation 52 nor 53 takes account
of the increase in discharge when the tail water rises above the
crest. If the air was shut out from the commencement, Q begins
to decrease as soon as the tail water begins to rise. See equation
46, page 94.
Bazin uses the simple weir formula q = ^GJ, 's/'^gH-^ (where C^ is
the inclusive co-efficient for the drowned weir and H^ the upstream
Q
head) and finds the ratio -^, C being the inclusive co-efficient for
the 'standard weir,' 3-72 feet high with a free fall and with the
same head 11^. His results are as follows : —
102
HYDRAULICS
fi
Ratio of
JJown-
stream
jj or Batio of Fall in Water to Height of Weir.
•05
■10
•16
■20
•20
■30
•85
■40
■45
•60
■60
■ro
E*
Head to
Height
of Weir.
Ratio 9^.
C
■0
l^Oo
ro5
105
ro5
1-05
1-05
1'05
\-05
1^05
105
1^05
ro5
106
■05
•84
■93
•96
•98
1^00
roi
roi
102
ro2
103
103
1-04
105
•10
■74
■85
•90
•94
•96
■97
•98
■%^
1 00
101
102
1 02
104
■15
•68
■80
•86
•90
•92
■94
•96
■97
•98
■99
1^00
l^Ol
103
■20
•64
•76
•82
•87
■90
■92
•94
■95
•96
■98
•99
100
102
■30
•58
•70
•77
•82
■86
■88
•90
■92
•94
■95
•98
•99
1 00
■40
■54
•66
■74
•79
■82
■85
■88
■90
•92
•93
•96
•98
•99
■60
■50
•61
■69
•74
■78
■81
•84
•87
•89
•90
•93
•96
•97
■80
•47
■58
■66
•71
•75
■79
•82
•84
•87
•89
•92
•94
•95
100
•45
•57
■64
•69
•74
•77
•80
•83
85
■87
•91
•94
•94
1^20
■44
•55
•63
•68
•72
•76
•79
•82
•84
■87
•90
•93
•93
1^50
■43
•54
■61
•67
•71
•75
•78
•81
•84
■86
•89
•92
•92
* This column shows -- when tlie tail water is below the crest, and the standing wave is
at a distance (art. 8).
Actually the ratio -^ is somewhat different with the weirs of
different heights for the same values of -^- and --^ , but the error
in the figure given is usually only 1 or 2 per cent., except for very
TT TT
small values of -^ and -^, and in these cases the ratio is alwrays
uncertain. The values 1-05 in the first line of the table agree
with the figure obtained by equation 46 (p. 94), when H., = 0. If,
for any given weir, G is supposed to be l^O, the abo\-e figures
show -?- for various values of // and //„. In this case, for a given
value of jy-, the figures are high Avhen // is high. This is due to
velocity of approach, the standard weir having been higli.
Bazin's figures may bo compared with those given on page 101.
Take for instance the casps where IT.. — 2 If.
If
(I
a
('.,
= -70
•50
•30
•10
•05
= M()
1-00
•GO
•20
•10
The figures on
p. 101 are •77
__ ■!.)■>
•67
•81
•7G
•74
and 71.
WEIRS 103
JT
Again for the case where B.„ = — .
o - 3
■45
IL
■70
I-L
•23
1-00
15
The figures on
05
■ p. 101 are -974
and -94:5.
96
•15
•98
In the above case, ■Vi^here ^ = '70, _J = 2^10 and—- or - — \— = ^-.
& G A Cr+ili 3"1
The excessive velocity of approach accounts for the high value
Bazin found that when H is reduced to about •166? or ■21G, the sheet,
instead of plunging beneath the surface (Fig. 75), suddenly assumes the
form shown in Pig. 80 (which he terms the ' undulating ' form, there being
generally waves near M), but this does not affect the co-efficient. If H
is now gradually increased, the undulating form remains till H is about
■28G or •29(t, but is unstable or liable at any moment to revert to the
other form.
14. Other Weirs. — The results of Bazin's observations on weirs
of other kinds are shown in the following table. Instead of
giving the co-efficient ratios Bazin gives the equivalent heads.
The conditions of flow are complicated in such cases, and formulss
can probably apply only with the co-efficient varying to a great
extent. The height II!^, to which the tail water can rise before it
begins to affect the discharge, varies greatly for different weirs.
For a weir in a thin wall it is very small, and it is largest foi'
weirs with flat tops. For the weir No. 5 in the table H\ was
fiTi. For weirs with a sharp top it was minus, zero, and plus
for downstream slopes of 1 to 1, 3 to 1, and 5 to 1 respectively,
the flat downstream slope in the last case having the same effect
as a large top width. For weirs with flat tops -66 foot wide, back
slopes varying from 2 to 1 to 5 to 1, H'^ is nearly -_', but when the
9 T-(
top was 1^32 feet wide R' ^ was —^■
o
The first two entries in the table on p. 105 show that with a flat-
topped weir c rapidly increased as H^ increased — Q being constant
— and became far higher than with a free weir. See table xix.; G in
a high weir differs little from c. When //j and H^ are both great,
as with a river in flood, much of the stream is not subject to con-
traction, Va approaches F, and C must be high, especially if the
front and back slopes are somewhat gradual, as is usual in such
weirs. Values of '80 to ^97 have been found, Q being, however,
merely calculated from the river section and slope, a difficulty which
may occur in such cases.
104
Eelative Heads on Submerged Weir.
-» —
'^
'~
Discharges per foot run of
Standard Weir in Thin Wall :
Refer
ence
Dimensions of the Weirs.
cubic metres per second.
•061 1 -110 1 •169 1 -310 1 ^480
i
Heads, //, on Standard Weir,
Remarks.
Num-
ber.
Down-
Top
widtl
Up.
s
metres.
stream
Slope.
in
me-
stream
Slope.
fl
35
is
•10 1 -16 1 -20 1 •SO 1 -40
tres.
[m
K
a
Corresponding Heads, Hi, on
the other Weirs, metres.
Weirs with sloping face or back.
■v
Verti-
cal.
E* -14
•18
■27
•36
1
1 tol
0-0
•75
-■OG -14
•19
•27
■36
ja^<niov
■06
•16
•20
■29
•38
the greater
■12
■18
•22
■31
•40
>discharges
■24
■35
•43
and when
//j is small.
E*
.16
•2]
■29
•37
o
1 tol
•2
J tol
■75
■12
■24
E-*
•17
■21
■27
■21
•29
■32
... 1
•39
•16
■31 1 -42 1
3
5 tol
Verti-
cal.
■75
•12
■24
■36
•17
•21
•27
•3.3 ...
■40 ^45
E*
■17
•22
■31
•41
4
5 tol
■2
J tol
■75
•12
■17
■22
■24
...
•33
•11
> Hi>H
■db
... 1 ...
... ^42
i
Weirs with flat top and vertical face
1
5
20
■75
and back.
■12
•24
•14
•18
••27
•35
■36
■12
•21
40
E*
•17
■29
•38
G
■75
•12
■14
•18
■22
•31 -39
•24
•28
■34 41
—
•36
in*
■12
•17
•41 1 ...
For small
■■'1 ■•'"i -T,' discharges
7
1
•35
•12
■13
•17
•22 30 39 i ^1 > ■^•
•24
; For greater
1
■36
■J 1 M discharges
E*
•11
•15
19
■27 -35 when /T, is'
8
•1
■'I'o
•12
•14
■18
31 -40 '■
small, and
\_
■24
•:;-) , ^43
H,> h\
K*
•11
•15
19 -27 -37
when H., is
1)
■1
■;?5
■12
■24
■36
•14
•17
•21
•33
•28 j ...
■36 ^40
... ^44
y
larger.
Tail water below crest and wave at a distance.
WEIKS
105
Hughes, adopting equation 51 with m=l, has worked out^ the
values of c for weirs Nos. 5 and 6 on the above list, and the results
condensed are as follows : —
Weir No. 5.
Weir No. 6.
Discharge in
cubic metres
per second.
^'i
ff2
H-i.
lU.
metres.
metres.
C
metres.
metres.
C
r
■122
•031
•50
•119
•000
•50
•061-
•122
•091
•70
•123
•090
•67
■161
•150
■87
•135
•120
•85
.
■163
•150
•74
■236
•151
•61
■216
•000
•56
•247
•211
■81
■219
•060
•56
•169-
•293
•271
■84
■220
•233
•277
•120
•ISO
•240
•63
•74
•72
•353
•242
•63
•301
•000
•61
•310 J
•360
•303
■80
•307
•120
•63
•396
•361
■88
•319
•210
•71
■413
•360
■72
•392 j
•409
•300
■68
•418
•360
■83
•439
•389
■84
r
•382
•000
■65
■480-
•384
•060
■63
•406
•240
■70
•442
•300
■68
< The effect of a submerged weir varies greatly according to the
state of the discharge. With low water it may act as a free weir,
and have great effect, for however small the discharge may be, the
upstream water-surface must be higher than the top of the weir.
With larger discharges the heading-up is less, and with a great
depth of water the weir may be almost imperceptible.
15, Contracted Channels and Weir-like Conditions. — Contracted
channels are (chap. ii. arts. 6 and 19) analogous to submerged weirs.
The co-efficients are generally not very well known. When an open
stream issues from a reservoir, or from a larger channel, or passes
' Madras Government Paper on Bazin's New Experiments on Flow over
Weirs.
106 HYDEAULICS
between contracted banks, or bridge abutments, or piers, c may
have any value from -60 to -95, being smallest when the angles of
the apertures are sharp and square (especially if there is a decrease
in section both vertically and laterally), greater if the angles are
chamfered or curved, and greatest when there are bell-mouths. The
co-efficients are also greater for large than for small openings.
The values for narrow openings are, roughly, for square piers, '6 ;
obtuse angled, '7 ; curved and acute, -8 to -9. For wider openings
add -1 or -2. The co-efficient may thus be 1-0 in a bell-mouthed
opening.
When a bridge or other obstruction in a stream has a waterway
less than that of the stream the real obstruction is frequently much
less than it seems to be. It is to be measured, not by the
difference between the waterway at the obstruction and that
upstream of it, but by the difference in the upstream and down-
stream water-levels. This is very often inconsiderable. A fall
of 1 foot gives a theoretical' velocity of 8 feet per second, and
•25 foot gives 4 feet per second. Bridges have sometimes been
altered or rebuilt owing to ' obstruction ' which was nearly harm-
less. Heading-up is most likely to be considerable with high
discharges, because the mean width of the channel is then increased,
while perhaps that of the contracted place is not. Thus the effect
varies in just the opposite manner to that of a submerged weir.
The real objection to a contraction is very often the expansion
which succeeds it and the eddies and scour which occur (chap. ii.
arts. 17 and 23, and chap. vii. art. 2).
Submerged weirs and lateral contractions are really varieties of
the same type of case, and some aspects of both of them will now
be considered.
A typical case of contraction is that caused bv bridge piers
(shown in plan and elevation in Fig. 80a). As in other cases, the
' drop down ' begins at AB where the reduction in the cross
section of the forward-moving water begins. It ends where the
section attains its minimum value. This is often about D, but it
may be at L if the surface slope DL is greater than the bed slope.
Below L the section again enlarges, and there may be a rise in the
surface or, if V is very high, a standing wave (chap. vii. art. 11).
Tt is pointed out by ITouk' in discussing floods in the Miami Valley,
' ( 'iilniliiUuii. of F/oir in Open C/imnic/n. Stato of Ohio, The Jliami Cou-
Bevvauc}' District, Tm'linical Reports, Pin-t iv. Dayton, Ohio, 1918.
WEIES
107
that Q is simply the discharge of an orifice of area BE under the
head AB — B being at the same level as D — with due addition for
velocity of approach, that the discharge of AB is not to be calculated
by the weir formula and added, and that such addition is based on
an erroneous principle, the error being due (a) to the absence of
a crest and (6) to the fact that the water treated as flowing over
a weir passes — downstream of the drop-down — through the area
treated as an orifice. But (a) does not seem to be a cause and (b)
is the same if there is a weir mn instead of piers. In any channel
which is locally contracted the formula — equation 16, art. 10,
vmmmmmm/^^mTmm
<:
Fig. 80a.
chap. ii. — for variable flow applies. The reasons for adopting the
weir formula are given in art. 2. The weir formula is fairly
correct for the case of a free weir in a thin wall (Fig. 65, art. 1),
or a notch (Figs. 82 and 83, art. 17). But directly there is any
sort of drowning (Figs. 66 and 68, art. 1, and Fig. 80, art. 13) com-
plications arise and a variety of co-efficients have to be used.
Only two levels are however required in most cases, namely those
of crest and upstream water.
The equation quoted expresses the principle that the fall in the
water surface, from A to L, less the loss of head from resistances,
is equal to the increase in the velocity head. This applies when
the weir or narrowing is bell-mouthed. It can also be applied to a
108 HYDRAULICS
sharp lateral contraction ^ if c„ is estimated — it generally differs
little from c — and the contracted area thus determined. See
chap. iii. art. 5. It is at the contracted area that the velocity
head must be taken. Something must be allowed for resistance
due to the eddies. Ordinarily the length BD is small and nothing
is allowed lor the friction in that length so that the first term on
the right ot the equation disappears. It is partly owing to this
and partly to the difiiculty in estimating the sectional area at L
(Fig. bO) that the equation is not usually applied to submerged
weirs. Equations 49 and 50 are used together. It is known that
part 01 the discharge comes from the section AB and part from
BK, but the theory is of course imperfect. It is known, however,
that both parts of the stream are contracted and pass at an
increased velocity through the reduced section at L. The water
y/////////////////////y//y^ ^
i
Fig. 80b,
^/////My/z/A
level at M is determined by the discharge of the downstream channel.
When a structure is being designed the water level at L is not
known, the probable level at M is considered and the weir formula
are used with such co-efficients as are found in practice to be
fairly accurate. When the amount of drowning is very great only
one equation is used.
In the cases dealt with in the report under reference the
distance BD was generally great — in many cases hundreds of feet, —
the stream contracting gradually. The loss of head from friction
in that length was separately computed and allowed for.
The formula was then
K, = ^2,7(7/+ -:,;") . . . (DSa)
-V '
wliere Kg is the velocity at DE and I^j that at AB. The value of
c was 10 unless there were angles such as to cause contraction.
' 111 this case tlio drop-ddwn begius at the commencement of the eddy which
replaces the pointed portion of the pier.
WEIKS 109
The total drop was generally from 1-5 feet to 4'8 feet, the velocity
6'7 to 23'1 feet per second, and the head lost in friction less than
25 per cent, of the total fall. The widths of most of the openings
are not stated, but in one case the width was 280 feet and the
sectional area at DE 7960 square feet. The depths were generally
great. The minimum sectional area was found by soundings and
measurements taken after the flood. If there were sharp edges or
square corners at the entrance a co-efficient of contraction was
applied. This was '7, '8, or '9.
Since the friction is approximately as F^, Houk considers that,
in allowing for friction in a considerable length in which the
velocity changes greatly, it is best to calculate the mean velocity
V not as 1^ 2 but as J 1 ^ 2 . Also— see Notes at end of
chapter vii. — a percentage should be added to the loss of head
Pig. 8O0.
because the square of the mean of the velocities in a cross-section
is less than the mean of the squares. In equation 16 and the
equations in chap. vii. art. 10, F is taken as ^"T — - and no per-
centage is added, but in these cases F^ and V^ differ only slightly
and no appreciable error results.
A weir-like condition, with water surface convex upwards, exists
wherever momentum is being imparted to the water, as when a
stream issues from a reservoir or from a larger stream (Fig. 80b),
or below a closed lock-gate when the water enters the lock from the
sides and flows along the lock. The case of a right-angled elbow is
similar. In all these cases the water has no previous momentum
in the new direction and the fall is approximately — - . F is usually
moderate and it is not necessary to calculate the fall in the water
surface, but it can often be seen. The case of a rapid with a
steep slope (Fig. 80c) is mentioned below.
110 HYDRAULICS
The statement (chap. ii. art. 11) that downstream of any abrupt
change in a uniform cliannel the flow is uniform is subject to the
above qualifications.
In the case of a thin-wall weir the air has access to the lower
side of the sheet and C, when H=\-^ feet and 6^=2-46 feet, is
about '66. In the case of a rapid (art. 1 1 and Fig. 80c) the air is
excluded, and when
S=lio\ 3 to 1 5 to 1 10 to 1
C is about -75 -65 -60 -56
S is the downstream slope. See Tables xviii. and xxii. With the
slope of 1 to 1 the drowning is slight. At about 3 to 1 its effect
— for the particular value of H quoted — counteracts that of the
exclusion of the air.
The value of the head, h, on the actual crest of a weir is of
interest in some cases, as will be seen. Bazin in his experiments
observed the head h (at left side of crest, Fig. 76, art. 10) v.ith
the following results, W being the crest width and H the head,
measured as usual to the right of the weir : —
H
W
10 1-4
4(Whenr= -66 ft.) = -94 -92 -91 "89 -87 -85
4 (When T'r= -33 ft.) =-97 -95 -93 -91 88 -85
A.(When 1F=-I64ft.)= -95 "93 -89 -86
xz
In the case of a rapid (Fig. 80c) Bazin foimd the following, S
being the slope of the rapid and h the head at the crest : —
H^ -33 -66 1 1-3 feet.
-^(When,S'=l in 5) -845 -863 -859 -852
^ (When, S-^,! in 10) -851 -869 -876 -873
When S was 1 in 5 and the crest width w as "66 feet instead of zero,
-jj. =-876 -891 -877 -871
WEIUS 111
For the greater heads the mean velocity of the stream, where
the head is h, is nearly the same as if AD was an orifice under a
head - and c^ = 1 . If at any point on tlie rapid tho velocity
exceeds the above, or the depth falls short of A, a standing wave
(chap. vii. art. 11) can occur. The velocity down the slope of a
smooth rapid may be very high. From N to M the surface may
be concave upward, as shown in Fig. 80c. Below this there is
uniformity of flow. In large rapids in India and Burma, in con-
nection with irrigation works, the slope is about 1 in 10 or 1 in 15,
the surface rough — boulder pitching, — and H much greater than in
the above experiments — say 3 to 1 1 feet. It is not known how the
ratio — . is affected, but it is possible that it is not very different
from the above. Observations are needed to decide the point. It
will then be possible to work out the depths further down the rapid
and to attend, in designing, to the question of the standing wave.
Section V. — Special Cases
16. Weirs with Sloping or Stepped Side-walls. — For a weir of
triangular section the formula is obtained by putting if, =0 and
4=Hn equation 36 (p. 71). Thus —
(3=xV n/S^ZS* ■ • (54).
Since I increases as H, in any triangular weir in which c does not
vary greatly, Q is nearly as f/*, that is, it varies much more
rapidly than with an ordinary weir. If two weirs, one triangular
and one rectangular, are so designed (Fig. 81) as to hold up the
water of a stream to a given level with ordinary supplies, the
triangular weir will allow floods to pass with a smaller head.
This applies to any weir with sloping sides. The triangular form
112
HYDRAULICS
is suitable for small drains. By making the sides of a weir at
any given level DE (Fig. 81) horizontal, and extending them
outwards, the rise of the water above DE can be limited.
Fia. 81.
The formula for the discharge of a trapezoidal weir (Fig. 82) is
Fig. S2.
obtained by putting II,=0 in equation 38 (p. 71). Thus —
Q=icj2^Hi{k+W-h)] . . . (55).
The quantity in the outer brackets is the crest length of the
equivalent ordinary weir. This length is less^than '"T ' because
the velocity of the water at the bottom of the section is greater
than at the top. If there is velocity of approach (II-\-nJt) must
be put for H in equation 55, or else put for c If r is the
ratio of the side slopes, that is, the ratio of ^4B to BC, then
AB
_— -=r=cota, AB=rH=Hooia, and I,— li='2iII~2H cot a.
Thus equation 55 may be written —
Q=IC j2^HHk+-mi} . . . (56).
17. Canal Notches. — A common problem on irrigation canals is
to design a weir so that the water-levels, CD, EF, etc. (Fig. 83),
upstream of it, corresponding to different discharges in the
channel of approach, shall be the same as they would have been
if the weir had not existed and the channel had continued uniform
and uninterrupted. If the cross-section of the channel of approach
is trapezoidal, the form of the aperture will be approximately
> In a triangular weir it la ^ I.
WEIKS
113
trapezoidal, and its crest will be at the bed-level of the canal.
Such a weir is termed a notch. It is usually, for convenience in
construction, built exactly trapezoidal and of the form shown in
Fig. 82, the lip being added to cause the falling water to spread
out and exert less effect
on the downstream floor.
In a large channel two
or more notches are built
side by side instead of one
very large notch. The
co-efficients, so far as
known, are given below.
If C is the same for
all heads the true theo-
retical form of the notch
is curved, the angles at C, F (Fig. 82) being rounded. The
slope of the sides is great for small depths because the co-efficient
for flow in channels increases rapidly for small depths ; but if C
increases fast with the head at small depths, as is highly probable,
judging from other weir co-efficients, the form is more nearly a
trapezoid. To design the notch, find Q and q, the discharges (or
the fractions of the discharges if there are to be several openings)
of the channel for two depths D and d. Then from equation 56
Fig. S3.
l^+-Srd=
.+ -8rD =
Therefore SriD-d)
\C,j2gd^ •
Q_
Q
(57).
(58).
<1
Or
_ lsj2g{C^Qdi-C^qlfi
■8xix2g{D-d)C\C,diD^'
i-2d,{D-d)C^C,d"-3 ' '
The depths d and D can be so selected as to make the notch
specially accurate for any given range of depth. In irrigation
canals (and still more in their distributaries) there is a certain
minimum depth, cZ^, below which the channel is not run. In such
a case it does not matter if the notch is inaccurate for depths less
than d^. To make its accuracy a maximum for depths between d^
and any greater depth, D^, the range of depth should be divided
114 HYDEAULICS
into four parts and the depths d and D taken at the quarter
points. Thus if
d=d,+^
4
4
If general accuracy is required over a range of depth from zero
to D,, then d—— and D= — ?. The formulae are, however, most
4 4
simple virhen D=1d. In this case equation 59 becomes —
_ C^Qdi-'2-9.29,C,qdi
^~4-28(fCA'^*x2-828^°
_ ^iS~2'828C2g /gQv
~ 12-10CA<^*
Substituting this value of r in 57
^^ g_ •8(C,C-2-828(7,g)
' fCi V2^(^* 12-10CiCj(i*
2-262C,g--8Cig+2-262(7,g
12-10CiC2<^t
2-262C',g--4C,Q
(61).
If Ci and Cj are each assumed to be equal to C,
_Q-2-828?
l1-lQGdi
(62).
. (63).
A A 1 2-262?- -40
And l,,— — r—
If it is desired to build a notch to the true form, that is not
strictly trapezoidal, the lower part corresponding to a small depth
in the channel may first be designed trapezoidal and the upper
parts designed in instalments, working upwards.
In deciding in which direction a notch is to deviate from the
true form, and for what water-levels accuracy is to be aimed at,
regard must be had to the special circumstances of the case. If
scour of the canal bed is feared or if there is difficulty, with low
supplies, in getting enough water into the distributaries, the
notch can be designed narrow.
If a notch is drowned its true form is modified. In Fig. 82 let
WEIRS 115
DE be the upstream water-level when the tail water is just level
with the crest CF. The portion C'BEF of the notch obviously
need not be altered. As the tail water rises above OF the
discharge through the notch ■ becomes gradually less than it
would be for a free notch with the same upstream water-level,
and the upper part of the notch must be widened as shown by
the dotted lines. In this case also a trapezoid can be drawn so
as to closely agree with the true form. As before, the trapezoid
can be designed so as to give nearly exact discharges for any
particular range of depths, or the notch can be designed to the
true form as above explained. The formulae for a drowned
notch are as follows : For an upstream depth d let g'j be the
discharge through ADEG and j'l through DGFE.
g=qi+<i'i_
= f 6', J2g{d - Iifl[l, + 2r/j + Srid - h)]
+ C\j2~g(d^{k+rh)h . . . (64).
For a greater discharge let D and II be the heights of AG and
DE above CF. Then
()=f C, j2^{D-I I)i[k+2rII+ •Sr(i)-/f)]
+ C\j2g{D-E){k+rH)II . . . (65).
If the upstream and downstream channels are similar in all
respects d—h=D—H and D—d=E—h. Let Z*=2d Then
d=D—d=H-h and H=d-{- k Therefore
<3=|C, ^j2^{d-h)i[k+2rH+-8r{d-h)]
+ C\^2g{d-h}{k+rir)E . . (66).
Subtracting 64 from 66 and putting C'l = C'a = 6' and 0'i = C'.2 = 0',
Q-q=^Oj2^{d-h)i[2rd]
+ C'j2g(d-h)[k{II-h)+r{E-h'')] . . . (67).
from which r can be found, and l^ can then be found from 65,
Q and (3i being selected at such depths as to make the trapezoid
most accurate at the points desired. If I) is not taken as 2d, or
if C\ and C^ differ, the equation will be complicated, and it may
be easiest to adopt the instalment process and design the notch
to the true curve, afterwards straightening it if necessary.
For notches having crest lengths of 2 to 6 feet c has been con-
sidered in India to be '65 to '73 and C "70 to '78, the figures being
116 I-IYDKAULICS
greater the larger the notch. Eecent figures given by Harvey ' are
as follows : —
11= 3 5 7 8 9 feet.
C= -848* -945* -95* -871 -91
It appears that the notches had been built too wide — perhaps
because was taken too low — and have since been narrowed.
18. Oblique and Special Weirs — If a weir is built obliquely
across a stream the discharge is that due to the full length of the
weir, provided the section of the stream passing over the weir is
small compared to that of the stream at the approach section.
In this case the water approaches the weir nearly at right angles.
Thus at low water the full length of the weir is utilised. A weir
AG (Fig. 83a) must be higher than BD in order to hold up low
^
G
Bi
E
^9
V. C
H
Fig. 83a.
water to the same level. But in floods the water passing over the
weir travels nearly parallel to the axis of the stream. AG probably
obstructs floods as much as BD does. If the low water discharge
is very small, the heights oi AG and BD may be almost equal and
the oblique weir may give a slight advantage in a flood. The
heavier the flood the less the advantage. The above remarks also
hold good if the oblique weir is V-shaped (GEF) and in whichever
direction the stream is flowing. If the channel is widened as per
dotted lines the full length GH is utilised even in floods, but if
v^ is high the gain as regards flood level— compared with the weir
BD — is not so great as when v^ is low. T^ie problem of construct-
ing a weir so that it will hold up low supplies and yet not form
a serious obstruction to floods can best be solved by means of gates
or shutters. See also cliap. vii. art. 8.
Circular weirs have boon usod whore there was not room for
straight weirs (Gourley, Min. Proc. Inst. G.E., vol. clxxxiv.), the
' I'rurrcdimis Punjab Engineering Congress, 1919.
• AG (Fig.'82) = 6 ft, C'i!'=2-63 ft, £C = 7-& ft.
t AOimg. 82)=5'4 ft. 0F=V6 ft., £C=9 ft.
■WHIRS 117
spigot ends of pipes — 6-inch to 24-inch — having been turned true
inside and outside and bevelled on the inside and the pipes placed
vertically with the spigot ends upwards and submerged, the water
thus flowing over the edges and into the pipes. The width of the
square edge above the bevel was -^ in. for the 6-inch and 9-inch
pipes and J in. for the others. A formula involving H^'^'^ was
arrived at and applies to heads up to 'ID. Calculated for the
usual weir formula the co-efficients are : —
Outside diameter {D) of pipe (inches) 6-9 10-08 13-7 19-4 25-9
c(whenH=-5ft.) -58 -58 '585 -59 -60
c(when//=-25ft.) -61 '61 -615 -62 -63
Each pipe stood in a square chamber whose diameter should be
32), the width of the channel of approach being 2D, baffle plates
being used to still the water and an air tube opening under the lip
of the weir.
Water has been made to flow up a pipe (Stewart and Longwell,
Trans. Am. Soc. G.E., vol. Isvii.) — of diameter {D) 2 to 12 inches —
and out at the top, which was turned true and bevelled on the
outside and had a sharp edge. Let H be the height of the water
above the edge. If II> 'ID there is a " jet condition " and
Q= 5-84 ZJ^-os //-"ss. If H< ID there is a " weir condition " and
Q = 8-8 i;i-29 H^.i^. Let £) = 1 and H= "1, than c in the usual weir
formula comes out 1 '02, the sheet probably clinging to the crest.
For smaller heads c is greater. If D = '5 then c is some 20 per
cent, less for the same values of II.
When the plane of a weir in a thin wall, instead of being vertical, is
inclined, the co-efl&cients can be obtained by multiplying that for a vertical
weir by a co-efEcient of correction Cj, whose value was found by Bazin to be
as follows : —
Inclination of plane of weir —
Upstream. Downstream.
1 to 1, § to 1, J to 1 ; vertical, J to 1, § to 1, 1 to 1, 2 to 1, 4 to 1.
Average value of Cj —
•93 -94 -96 rO 1-04 1-07 110 1-12 109.
The heights of the weirs when vertical were 372 feet, 1 -64 feet, and 1 -15 feet.
The co-efficient is a maximum when the weir is inclined downstream at 2 to
1, that is, when the height of the crest above the bed is half the distance
of the crest downstream from the base of the weir. The weirs were without
end contractions, and the head ranged in each case from about '33 feet to
1 -48 feet.
118 HYDRAULICS
Examples
Example 1. — A weir in a thin wall is 25 feet long and 3 feet
high, and H is 1 foot. The channel of approach is 30 feet wide.
Find Q.
The crest contraction is complete, and the end contraction so
nearly complete that no allowance need be made for it. From
table xiv. c is probably -612. From table xii..£'=3-275. Then
0' = 25x3-275 = 81-88 cubic feet per second.
To allow for v by the usual method, ^ = 30x4 = 1 20 square feet.
Let Q be assumed to be, say, 84 cubic feet per second. Then
w=T%V='70. From table i. /i=-.0076. Let n=l-^. Then
nh=-Q\Ql, H+nh=\'Q\Q. The corresponding correction in
(H+nhy and in Q' is 1'5 per cent., and Q is thus 83-11 cubic feet
per second.
To allow for v by table xiii. — = ^l^=4:-8. When c is -60
^ a 25x1
c„ is about 1-015. When c is -61 c„ is about 1-016. This makes
Q=83-12 cubic feet per second.
Example 2. — A river 50 feet wide has a maximum discharge of
600 cubic feet per second, the depth being then 3 feet. A weir
with a rounded crest (c=-80) is to be built in the river so as to
raise the flood level by 1 foot. What must be the height of the
crest above the bed ?
The discharge, q, per foot run of weir is 12 cubic feet per second,
and table xii. for c=-80 gives /i=4-28. Therefore
{H+nh)i=~ =■2-80. From table xi. H+nh=l-9Q feet. But
4-28
«)=3-0, and h (table i.) = -14 foot. Therefore, n being 1-0, -ffis
1-85 feet, and the crest must be 2-15 feet above the bed. The
result is quite accurate, supposing that the channel downstream of
the weir is altered for a long distance so as to give a free fall over
the weir. Otherwise the weir will be drowned, .ff. being -85 foot,
but judging from Bazin's results (art. 14) with weirs having a
moderate top width and sloping back and face, the discharge will
hardly be affected, 11^ being ohly •461f,. Actually // would
perhaps be 1-9 or 1-95 feet.
Example 3.— A river whose moan width is 50 feet, depth 10 feet,
and moan velocity 3 feet per second, has a bridge built across it.
The piors and abutments are square, and the total width of the
"WEIKS 119
water-way in the bridge is 30 feet. Find the heading-up caused
by the bridge.
Let c be -60. Since Q is 1500 cubic feet per second, and a is
300, therefore F'=--l^'^„-- = 8-33 feet per second. From table i.
300 X -60 ^
JT=1'08 feet nearly, but as there is high velocity of approach
H will be less, say 1 -0 foot. Therefore
^ = 50 X 11 '0 = 550 square feet, and v=i/J>^-=2-73 feet per second.
From table i. A= -1 16. Let «,= 1 -0. Then E+7ih=l 1 16. From
table i. V=S-i'l feet per second, which is too great by nearly
2 per cent., and H is therefore less than 1 foot by 4 per cent.,
that is, it is -96 foot.
Example 4. — The depth of full supply in a canal is 5 feet. The
discharges with depths of 4 feet and 2 feet are 153 cubic feet and
46 cubic feet per second respectively. Design a trapezoidal notch
for a free fall in the canal. The co-eflficient is '66.
From equation 62, page 109,
._ 153-2-828x46 ^
12-10x-66x2*'
From equation 63, page 109,
^_ 2-262x4-6--4xl53 _
6-05 X -66x2*
' = ■51.
:3'78 feet.
Example 5. — A weir in a thin wall is 4 feet high and E is
1 foot. The bed of the stream becomes filled up, so that the
depth above the weir becomes 2"5 feet instead of 5 feet, but Q is
unaltered. How is H affected ?
A
The ratios — are 5 and 2'5 nearly. From table xiii., r, being
•60 and n being 1'33, the values of c„ are 1-013 and 1'057, so that
Q is increased about 4-4 per cent, if //is the same. H will therefore
be less than before by |x4-4 per cent., that is, it will be -97 feet
120
HYDRAULICS
Table XL
Values of H and JJ*. (Art. 1.)
H
Hi
Din-.
01 H
n
h2
Diff.
■01 //
H
jfi
Diff.
-01 /f
■04
•0080
■0032
■60
•4648
-0119
1-8
2-415
■0202
■05
•0112
■0035
■62
•4882
■0121
1-86
2-616
•0205
•06
•014?
0038
■64
•5120
■0123
190
2-619
-0208
•07
•0185
■0041
■66
■5362
■0125
195
2^723
■0210
•08
■0226
■0044
■68
■5607
■0127
2-
2^828
■0214
•09
•0270
■0047
•70
■5857
■0129
2 05
2935
-0216
•10
■0316
■0049
•72
■6109
■0130
2^1
3043
-0218
•u
•0365
0051
■74
■6366
■0131
215
3152
0221
•12
■0416
0053
■76
■6626
■0132
2^2
3263
■0224
•13
■0469
•0055
•78
■6889
•0133
2-25
3-375
■0226
•14
•0524
0057
■80
■7155
•0135
2-3
3-488
•0228
•15
•0681
0059
■82
■7426
•0137
2-35
3-602
•0231
•16
•0640
0061
■84
■7699
•0138
2-4
3-718
•0234
•17
•0701
0063
■86
■7975
-0140
2-46
3^834
•0237
•18
•0764
0064
■88
■8255
■0142
2-5
3953
0238
•19
•0828
0066
•90
■8538
■0143
255
4^072
•0240
•20
•0894
0068
•92
■8824
■0145
2 6
4192
•0242
•22
•1032
0072
•94
■9114
■0146
2-65
4314
0244
•24
•1176
0075
•96
■9406
■0148
2-7
4-437
•0246
•26
•1326
0078
•98
■9702
■0149
2-75
4-560
•0250
•28
■1482
0081
1-0
1^000
■0152
2-8
4685
•0262
•30
•1643
0084
ro6
1^076
■0156
2-85
4-811
0254
•32
•1810
0087
1-ip
1-154
-0158
2-90
4939
•0255
•34
•1983
0089
M5
1233
•0163
2-95
5066
■0260
•36
•2160
0091
1^2
1315
■0166
30
5196
-0262
■38
•2342
0094
1-26
1398
■0168
31
5^46S
0266
•40
•2530
0096
13
r482
■0172
32
5 -7 '24
-0271
•42
•2722
0099
136
r568
■0176
3 3
5-995
-0275
•44
•2919
0101
14
1^657
■0178
34
6-269
•0279
•46
•3120
0103
1^46
1-746
•0182
3-5
6-648
•0283
•48
•3326
0106
^5
r837
■0186
3-6
6-831
•0287
•60
•3536
0109
r55
r930
■0188
37
7-117
■0291
•62
•3750
0112
16
2024
■0190
3-S
7-408
■0294
•54
■3968
0113
1^65
2.119
■0104
3-9
7^702
•0298
•56
■4191
0116
1-7
2-217
•0197
4-0
8-000
•0302
•58
■4417
0117
1^75
2-315
•0200
WEIRS
121
Table XII.— Values of K or ^cjig or 5-35c. (Art. 1.)
c
K
C
K
c
K
■001
•00535
61
3^264
•81
4-334
■002
■01O7
62
3^317
■82
4-387
■003
•01605
63
3371
•83
4-441
■004
•0214
64
3^424
•84
4^494
■005
•0268
65
3^478
•86
4-548
■006
•0321
66
3-531
•86
4^601
■007
•0375
G7
3^581
•87
4-655
•008
•0428
68
3^638
•88
4-708
•009
•0482
69
$•692
-89
4^762
•5
2^675
7
3^745
•9
4^8 15
•51
2^729
71
3799
•91
4^869
•52
2^782
72
3^852
•92
4^922
■53
2-836
73
3906
•93
4^976
•54
2-889
74
3959
•94
5^029
•55
2^943
75
4^013
■95
5^083
•56
2-996
76
4-066
•96
5^136
•57
3^050
77
4^120
•97
5-190
•58
3^103
78
4^173
-98
5-243
•59
3^157
79
4^227
•99
5-297
■6
321
8
4^28
1
5-35
Table XIII. — Co-efficients of Correction, c„,
FOR Velocity of Approach. (Art. 5.)
A
c = -60
c=-80
Values of n.
Values of n.
1-4
1-33
1
1-4
1-38
1
2
r098
1*093
1-067
1-198
1-189
1-129
2^2
1^079
1^075
1-055
1-156
1-149
1-105
2^5
1060
1^057
1^042
1-115
1-110
1-079
3
1^041
1^039
r028
1074
1-071
1-050
4
1^022
1021
1-015
1-041
1-039
1-028
5
roi4
roi3
roo9
1-025
1-024
1-017
7
1^007
1^007
1-005
1^012
1-011
1-008
]0
roo3
roos
1-001
rooe
1-U06
1-004
122
HYDRAULICS
Tables XIV. and XV. — Co-efficients of Discharge, c, foe
Weirs in Thin Walls with Complete Contraction.
(Art. 6.)
XIV. — Ordinary Weirs.
Head
Length of Weir ir
Feet.
Feet.
•6
1
2
3
6
10
19
•1
■652
■653
■655
•656
•15
•598
■605
■630
■638
■640
•641
•642
•2
•593
■600
■623
■630
■631
•633
•634
•25
•583
■595
•617
•624
•626
•628
•629
•3
■578
•593
•612
•619
■621
•624
•625
•4
•578
•591
•607
•613
■615
•618
•620
■5
•582
■589
•602
•608
■611
•615
•617
•6
•584
•587
•598
•605
■608
•613
•615
•7
•585
•585
•594
•603
■606
•612
■614
•8
•588
•584
•590
•600
■604
•611
-■613
■9
•590
■584
•587
•598
■603
•609
■612
1
■592
•583
•585
•595
■601
•608
■611
1-2
■591
•597
•605
■610
1^4
•573
■587
■594
■602
■609
1-6
•571
■582
■591
■600
•607
17
...
•599
•607
2
...
■576
...
XF.— Short Weirs.
Head
in
Feet.
Length of Weir ii
Feet.
i
•033
■066
■0!)9
■164
■246
•329
•654
•03
•634
•05
■620
...
•618
•10
•605
•608
•618
•624
•13
■613
■605
■605
■618
•16
•629
•614
•604
■598
•611
•25
•653
•(12S
•or2
■ll(l^2
■594
•33
•048
■027
•012
•591
•39
•079
■(i-tr>
■627
■Olii
•589
■590
•66
•668
■0411
■028
■614
■593
•591
.80
■066
•642
0^:8
•615
■594
WEIRS
123
Table XVI. — Co-efficients of Discharge, c, for Weirs in
Thin Walls without End Contractions, but with Full
Crest Contraction. (Art. 6.)
Head
in
Length c
f Weir in Feet.
1-6 to 6-6
2(?)
3{?)
4
5
7
10
15
19
Bazin's
Co-
efficients.
Smith's Co-efficients.
•1
•659
•658
•658
■657
•657
■15
•652
•649
■647
■645
•645
•644
•644
•643
•2
•662
•645
■642
■641
■638
•637
■637
■636
•635
•25
•655
•641
•638
■636
■634
•633
•632
631
•630
■3
•652
•639
•636
■633
•631
■629
■628
■627
•626
■i
•646
•636
•633
■630
■628
■625
■623
■622
•621
■5
•640
■637
•633
■6:i0
■627
■624
■621
•620
•619
•6
•637
•638
•634
■630
■627
■623
■620
■619
■618
•7
•635
■640
•635
■631
■628
■624
•620
■619
■618
■8
•633
■643
•637
■633
■629
•625
•621
■620
■618
■9
•633
■645
■639
■635
■631
■627
•622
■620
■619
1
•632
•648
■641
■6.37
■633
•628
■624
■621
■619
1-2
•631
■646
■641
■636
•632
•626
•623
■620
1-4
•630
■644
■640
■634
•629
•625
■622
1-6
•627
■647
•642
■637
•631
•626
■623
1-7
•626
•638
•632
•626
■623
1-8
•625
...
Table XVII. — Corrections for Wide Crests. (Art. 10.)
(The correction is always minus except when marked plus. )
Head
in
Width of Crest in Inches.
Feet.
1
2
3
4
6
s
10
12
24
•10
■007
■016
•018
■018
■017
•017
•017
•017
•017
•15
-f002
■017
■023
■024
■025
•025
•025
•025
■026
•20
•012
•024
029
■031
■032
•033
■033
■034
•30
+ •005
•017
■030
•041
■045
•047
■048
■050
•40
■010
■022
■045
•055
■060
•062
•066
•45
H-^oog
...
•50
■006
■041
•060
•069
■074
•082
•60
■031
•059
•075
■083
■097
•70
•017
■052
•075
•089
•112
•80
•000
•040
•071
•091
■125
•90
-I-019
■027
•062
■089
■137
1^0
■056
•050
•082
■149
1^2
+ ■025
■021
■061
■168
1^4
+ 013
■032
-180
r5
■015
124
HYDRAULICS
Tables XVIII. to XXII.— Inclusive Co-efficients, C, foe
Weirs 6-56 Feet Long without End Contractions.
XVIIL— Weirs
in Thin JValls.
(Art. 6.
)
Height
of Weir
Head in Feet.
1 1
1
in Feet.
•164
•23
•83
■63
•06
■720
■82
•98
1^16
1^31
1-48
VM
1^80
r97
•66
•687
•683
•689
■703
•735
•750
■98
•680
•672
■671
■677
■689
■700
■713
■723
•734
■743
...
1-31
■677
■668
■663
■665
■671
■680
•690
•699
•708
■716
■723
■729
■735
1-64
•675
665
•659
•657
■660
■668
•675
■683
■689
■696
■712
■709
■714
1-97
■674
■663
■656
•653
■654
■659
■665
■671
■677
■683
■689
•693
■699
2-62
■674
•662
■653
■647
■647
■648
■651
■657
■660
■664
■668
■671
■677
3-28
■674
■660
■651
■644
■642
■643
■645
■648
■650
■653
•656
■658
■662
4-92
■672
■660
■650
■641
■638
■636
•636
■636
■636
■638
■639
■640
•641
6-56
•672
■659
■650
•641
■635
■633
■632
632
■632
632
632
■632
■632
XIX. — Weirs mth Flat Tops and Vertical Face and Back.
(Art. 10.)
Dimensions
of Weirs.
Heads in Feet.
■Width
in Feet.
Height
in Feet.
■3
■7
l^O
1^2
14
1-6 1 2-0
2-3
3-0
3-8
4-3
•50
5*0
e-o
16^ 30
11^25
12^24
11^25
>••
■49
•50
...
...
6^56
4-57
*••
• ••
•45
•48
■51 (t)
6 ■561
2^46
•45
■48
•48
■50
...
5 •88
11 ^25
• ••
>•*
■50
3^17
11^25
•51
■52
■54
2-62
4^57
■64
■65(?)
2^62i
2 •46
•48
■49
■50
•62
...
1^65
ir25
...
■56
■62
1
1^31
2-46
■60
■51
■54
■h9
—
' ■\Vlien the width ■was in-
creased by •83 feet and the
upstream edge rounded to a
•66
2^46
■52
•58
•66
.1.
•70
radiiis of •SS feet G ■was in-
•66
M5
■■58
■60
■67
...
• ••
creased 10 to 14 per cent.
•48
11^25
246
...
■62
■62
•83
■57
■fiR
■m*
•71*
♦ Thea
5 are for sheets drowned under-
•88
115
■57
■72
■IT
neath. All other flgiires are for free sheets,
and the corresponding figures for Bazin'a
•18
246
•bS
■80"
•Tl"
weirs for sheets depressed or drowned
•16
1'15
•69
■75*
■78"
underneath are the same to within, gener-
ally, 4 per cent.
WEIES
XX. — Weirs ivith Rounded Topi. (Art. 12.)
125
Sections of Weirs.
Dimensious
oJ Weirs.
Head in feet.
1
Kadius of Crest.
Height
in Feet.
•3
•7
Vi
■9
2-5
4-7
Kg. 69, p. 82,
•34 ft.
upstream,
•40 ft.
downstream
1'64
•67
•79
•86
Kg. 78, p. 99, .
•26 ft.
1-64
•72
•84
•84
...
lig. 78a, p. 99, .
3 •37 ft.
5-3
1-64
•57
•57(?)
•59
•62(?)
•65
•61
■64
•675
Fig. 78b, p. 99,
When the height -was 8 feet and radins of curvature 10 feet, c
was -60 when H=2 feet. When weir raised by laying a I'xl'
timber along the crest, c was •es when £f = 2 feet or 2-7 feet
(Horton, op. cit.).
XXI. — Weirs with Steep HacJcslopes. (Art. 11.)
2^
Back Vertical.
Back J to 1.
Back f to 1.
n
Slope of
Ji'ace of
Weir.
Head in Feet.
Head in Feet.
Head in Feet.
ft-S
a.M
1
ii^
S
•3
•7
Vi
1^2
5^0
•3
•7
l-i
•3
•7
1-4
1^64
Vertical
■65
•76
•71
•75
■78
•71
0^00
1^64
itol
•68
•78
•74
1^64
2tol
•75
•79
•77
■56
•73
•72
•56
•73
•71
1^64
Vertical
•33
1-64
1 to 1
•59
•72
■80
■57
•73
•78
•60
•73
•80
1-64
2 to 1
•61
•71
•77
4^7
2tol
•70
•67
r64
2tol
•58
•65
•73
•66
4^9
2tol
•605
49
3 tol
•67
•67
4^9
5tol
•63
•63
126 HYDRAULICS
XXII. — Wews with Flat Badc-slojjes. ( Art. 1 1 . )
,►1
li'iujo Vortical.
Face 1 to 1.' Face 2 to 1.
Slopo of
Back ot
ft'E
■ss
1 1
Head in
Feet.
Head in
Feet.
Head in Feet.
Woir.
fo.S
H^
H
■7
1-4
•3
■81
•7
•80
•76
•3
•79
•7
1^4
2 3
4
6
r
1-64
•78
•77
00 ]
2^46
•72
■73
75
1 to 1
1
4-9
•79(?)
•77(?)
•73(?)
■70(?)
•68(?)
•33
1^64
■57
■71
■78
60
•72
82
•61
■71
■79
■66
1-64
•58
■65
74
f
1-M
•65
■69
■72
■72
•75
•79
•73
•74
78
■<'''\246
65
■66
■69
■33
104
■56
•66
•72
•61
•69
■78
2 to 1
f
164
■58
•63
■74
•58
64
■73
•66-^
2^46
4^9
48
•56
•68
■5 J
•61
■71
•55
62
•63(?)
■70
■66(?)
■68(?)
•69(?)
69(?)
1-31
2-46
164
•49
■55
51
■64
•58
68
—
—
—
-— --■
3 to 1
■00
•
i
Heads.
1
1-7 3^3 4-8 1
•00 1
1-64
66
•66
■69 68
•69
■71
2-46
■58
■58
•00
1
;-) to 1
•33/
1 64
•54
58
63
56
•64
■70 -58 -65
69
4-9
■5 [•66{?)
68(?)
•67 ;64
•67
•66
1-64
•58
62
•68
j
10 to 1
•00
2-46
■52
54
•56
1
i
For a weir 8 feet high, with upstream slope 5 to 4 and down-
stream slope 1 in 6, c was "69 when //=1'9 feet. When the weir
was raised by laying a 1' x 1' timber along the crest, c was "68
when H= 1-2 feet (Horton, op. cit.).
For rapids, C has been fovmd to be '65 to -67, the face having a
slope of 1 to 1, 7/ being 2'5 to 4'2 feet and Va being 2 to 3 feet per
second {The Control of Water, Parker).
CHAPTEE V
PIPES
[For preliminary information see chapter ii. articles 8 to 21]
Section I. — Uniform Flow
1. General Information. — In a uniform pipe, AB (Fig. 84), let the
length ^C, amounting to two or three times the diameter, be termed
the mouthpiece of the pipe. At the entrance of the pipe a head — -
must be spent in imparting momentum to the water. This causes
a loss of pressure head only, and not of total head. In exchange
for the loss of pressure the water obtains a velocity head — — , but
this is finally lost in the receiving reservoir, where the energy
possessed by the water is wasted, in eddies. There is also a loss
in the mouthpiece depending on the co-efficient of resistance
(chap. iii. art. 6), and varying from about "06— - in a bell-
es'
127
128 HYDRAULICS
V ■ ■ ■ ■
mouthed, to aboiit -50-^ in a cylindrical mouthpiece. This
last occurs if the pipe simply stops short flush with the side of the
reservoir without being splayed out. If the pipe projects into
the reservoir, and ends without a flange, the loss of head is about
•93 The total loss of pressure head at the entrance of a pipe
is thus (l+«„) — where «^ varies from -06 to -93. This loss of
head is the height FI). The line of hydraulic gradient is FEG.
In equal lengths i,, L^, etc., the falls in the line of gradient or
losses of head by friction are equal. If the inclination of the pipe
is uniform, as in A'B', the line of virtual slope is straight, but not
otherwise. Generally, however, the variations in the inclination
of the pipe in lengths L^, L^, etc., are not enough to cause great
differences in the lengths of their horizontal projections, and the
line of virtual slope is practically straight. Generally the length
of a pipe is so great that the loss of head at the entrance may be
neglected in estimating H, and the length of the mouthpiece
in estimating L. S is then found more easily. The actual posi-
tion of the pipe is of no consequence. The virtual slopes and
discharges of the pipes AB, A'B', etc., are all equal, provided
the roughnesses, diameters, and lengths are equal. If the pipe
discharges freely into air, the virtual slope is FB. Pipes are
always assumed to be circular in section unless the contrary is
stated.
If at any point R the line of the pipe rises above the line of
the hydraulic gradient, the pressure is less than the atmospheric
pressure. At such a point air may be disengaged from the water
and the flow impeded, the line of gradient being shifted to FB
(loss of head at entrance not considered) and the pipe BX running
only partly full. If the height MR is more than Si feet and
R is lower than X, flow is still possible.^ The above refers
to cases in which the water is subjected throughout to ordinary
atmospheric pressure. If the pressures on the two reservoirs are
unequal the heads must be calculated (chap. ii. art. 1) and the
gradient xy drawn accordingly. Arrangements must be made for
periodically drawing off the air which accumulates at 'summits'
such as 7i lying above the gradient line.
With small pipes a great increase in the temperature of the
water increases the discharge. The following results have been
found : —
' See Notes at end of chapter.
FT PES
129
Diameter of
Pipe.
Increase in Teniperatnrc
of WaLcr.
Increase of Discharf^c.
From
To
Inches.
1
1-5
2
60°
57°
52°
212°
120°
59°
25 per cent.
8 per cent. (F about 8'5).
10 per cent. { V about 5'7).
Discbarge was perceptibly-
increased.
The pressure in a pipe, after allowing for difference in head,
decreases somewhat in going from the circumference to the centre.
Let D be the diameter of a pipe. Then B is -r or half the
actual radius. Since the sectional area is as D°, ;JB as JJ), and
since C also increases with D, the discharge increases more rapidly
than DK If two pipes are nearly equal in diameter, their
discharges will be nearly as D^. Allowing for increase of C, a
pipe of 2 feet diameter will discharge nearly as much as six pipes
of 1 foot diameter. To double the discharge of a pipe it is only
necessary to increase the diameter by about 30 per cent. Since
/^increases as JS, and C also increases slightly with S, the dis-
charge increases rather more rapidly than JS. In order to double
the discharge S must be more than trebled. Doubling the slope
increases the discharge by perhaps 50 per cent. For a given
head H the slope is inversely as L, and Q therefore increases
more rapidly than — rj. It is clear that slight errors in measuring
the diameter of a pipe, or an insufficient number of measurements
when the diameter varies — as it nearly always does — may cause
considerable errors in discharges or co-efficients.
All the ordinary problems connected with flow in uniform pipes
can be solved by means of equations 14 and 15 (p. 21), some
directly and some by the tentative process. The problems referred
to are those in which one of the quantities Q, S and I> has to be
found, the others being given. V can, of course, always be found
from D and Q without difficulty, or either of those quantities from
Fand the other. Pipes are generally manufactured of certain fixed
sizes, and when the theoretical diameter has been calculated the
most suitable of these sizes can be adopted, unless a special size
I
130
HYDRAULICS
is to be made. To facilitate calculations various tables have been
prepared. The method of using them and of dealing with the
above problems will be clear from the examples given and the
remarks which precede them.
2. Short Pipes. — When the length of a pipe is not very great
the velocity may be high, the co-efficient C may be outside the
range of experimental data, and its value then can only be
estimated. For cases in which L is not more than lOOD the pipe
may be treated as a short tube, and equation 7 (p. 13) used.
The following values of c have been found : —
Mateeials and Diambtbes in Inches.
Katio of
Iron.!
Cast Iron.'
Earthen-
ware."
Cement or Stoneware." ,vS*4
LioD.
1
4 6
8
10
6
4
6
9
12
18
■74
24 •6'x-4'. '
6
8
■ 'r-r
12
■71
15
•68
21-6
•S4
24
•73
■80
•64
27
•76
•66
36
•68
•71
■69
■60
•65
43
•74
•63
43
64
60
•63
•60
•63
•66
•48
Kull details uf the expeniiients are "tot kiionn. Wlien
72
■m
Fis not too high it is best to adopt the usual formula
100
•65
fur pipes. See example 4 at end of chapter.
108
•60
For the brickwoik pipe the ratio of i to 7) has re-
ference to the '6' dimension.
» Fanning (H-=2-36 ft.).
^J5gypt.an Irrigation Experiments^ p„„^.„j £,^„eeri»9
"Punjab do. H=3'to3ft.). Co"9re«, 1916 (P.R
• do. do. {H= 3 to 3 ft.). '•\''m>»^) All the
» Madras Kistca Division E.xperi- P>P«s were sub-
ments (S = 25 to 32 ft.), merged.
All the experiments were made with small heads. The shorter
the pipe the greater the proportionate loss of head at the entrance
and the less the variation of c for a proportionate increase in L.
Thus when L increases from 25i to SOi. c does not decrease so
much as when L increases from 50i to lOOL.
3. Combinations of Pipes. — If a pipe does not simply connect
two reservoirs, but is, say, a branch supplied from a larger pipe
and itself bifurcating, its discharge can .onlj' be ascertained by
tapping it and attaching pressure columns.
When a wdter-main gives off branches it may undergo reduc-
tions in diameter. Suppose that the conditions in such a main
are to bo determined when no water is being drawn off by the
branches. If the discharge of the main is known the loss of head
and gradient in each length can be found. Suppose, however,
that only the total loss of head H is known. Obviously the
PIPES 131
gradient in any length will be flatter as D is greater, and JS will
be roughly as — - or -^ as -:f- or -H" as ^^. Thus if the total loss
of head is known the loss in each length can be roughly found,
the gradient being sketched and the discharge computed.
When greater accuracy is required let D' be an approximation to
the average diameter of the whole main. With this diameter
and gradient — find an approximate discharge Q', and thence the
velocities V-^, V„, etc. Then for any length L^, Cj JRi= -,\-
The slopes S^^, S^, etc., can then be found, and the losses of head
are L^ S-^, L^ S^, etc. If these when added together are not equal
to H the discharge Q' must be corrected. When Q has been
found accurately the diameter D of the equivalent uniform main
is known. It is such as gives the discharge Q with the gradient
TT
-=^. If the above problem again occurs with the same pipe, but a
Ij
different value of H, there will be no difficulty, for D will be
practically unaltered.
Let Fig. 85 represent a main of uniform
diameter, and let its discharge be drawn
off gradually by branches. If the dis-
charges at 31 and N are Q and zero
respectively, and if the discharge is
supposed to decrease uniformly along
the whole length of the pipe, then the
line of gradient will be a curve. If x and ?/ are the ordinates
of any point in the curve, and ^i and B are constants, Q=Ax.
But if C is supposed constant, Qz=B JS=B (j~) • Therefore
li=P'- Integrating, 2/= ^.^
When x=L, y-^= — -,. i^andthe mean gradient jZj= ^ .i^ But
SB Li ox»"
when x=L, ^^ is -=-; .U, or the mean gradient is one-third of the
ax B'
gradient at M. The total loss of head is one-third of what it
would have been if the whole discharge Q had been delivered at
N. As C increases with S the fraction is really greater than one-
third.
132
HYDRAULICS
If in a branched pipe (Fig. 86) the pressures at J, B, are
linown, the discharges can be found by assuming a pressure head,
//, at D, and calculating the discharges Q^, Q^, Qs- If Qi does not
o
equal Q^-{- Q^, then // must be altered and a fresh trial made. Q.,
may be plus, zero, or minus according to the direction in which
the water flows.
Let E (Fig. 87) be a water-main, EF a branch, and GK a
pressure column, and let there be a three-way cock at G. If no
water is being drawn off at F the water rises to a height K,
determined by the pressure in the main, whether GK or GF is
open; but if water is being drawn off at F the height GK^fill be
less when GF is open. If EF is a house service-pipe and GK a
pipe rising to the ground-level outside the house, then by means
of a pressure-gauge at K an inspector can tell, without entering
the house, whether water is being used in it or not.
In a system of bifurcating pipes (Fig. 88) such as that used for
the water-supply of a town, the pressure heads sufficient to force the
water to the required levels
at various points, L, K, F,
having been determined, the
gradients corresponding to
o imaginary pressure columns
at these points can be drawn,
■'^ and the required discharges
'7i> !?ai etc., being known, the
diameters of the various pipes
can be calculated. Suppose the system to be at work, then if the
consumption in a branch FG is increased, the pressure head at F
will 1)0 lowered and the branch FH will not be able to obtain its
Fio. 88,
PIPES 133
estimated supply, unless its conditions are similar to those of FG.
The lowering of the pressure at F causes an increased discharge
in LF, and a lowering at L, and thus more water is drawn in from
the reservoir, but not to the same amount as the increase taken
by FG. Thus any excessive consumption tends to partially
remedy itself, firstly by preventing water being forced to high
levels in its neighbourhood, and secondly, by drawing more water
into the main. (Cf. chap. vii. art. 6.)
4. Bends. — The loss of head, iZjj, " due to a bend " in a pipe, is
the loss over and above the loss, H, from friction in the same length
of straight pipe. It is usually put in the form Z^, -— . With a
view to ascertaining the values of Z^, for bends of 90° in pipes of
diameters ranging from about 2 inches to 2'5 feet, experiments have
been made by Weisbach,^ Williams, Hubbell and Fenkell,^ Schoder,^
Davis,^ and Brightmore.* The bends experimented on had radii {R)
of 2'5Z> to 242), D being the diameter of the pipe. A detailed
review of all the experiments is given in The Engineer, 26th May
1911. The general result is roughly that Zj^ in a 90° bend may be
about '10 to -40, and that the loss of head Hs is generally only a
fraction of H.
The experiments show that great care is needed to ascertain H^-
The difficulty is to determine what the loss would have been in a
straight pipe. A small error in ascertaining this upsets the cal-
culations completely. It is essential that the diameter and con-
dition of the bend should be the same as in the tangents, and the
same as in the straight length with which the bend is to be com-
pared, and that the pressure columns should be so placed that they
are not affected by disturbances due to the bend itself, or to any
other bend or any other cause operating upstream. A length of
100 pipe diameters is perhaps necessary to let a disturbance die
away. These conditions have not been completely fulfilled in any
of the experiments. Owing to the smallness of Hs its actual value
has been obscured by the errors, and the results of the experiments
are generally considered to be unreliable. Details of them are,
however, given below.
When B is great the resistance per foot run of pipe is small, but
the length is great, and this may cause a fairly high value of He.
As bearing on this point it may be observed that views are dis-
crepant as to the effect of a very slight change in direction.
* Mechanics of Engineering. ^ Trans. Am. Soc. C.E., vol. xlvii.
^ Trans. Am. Soc. G.E., vol. Ixii. ' Min. Proc. Inst. G.E., vol. clxix.
n I uivA u uiL^o
Williams, Hubbell and Fenkell state that a divergence from the
straight of 2° had considerable eifect. Schoder found that C, for a
pipe laid not strictly straight, i.e. with a slight zig-zag appearance,
was the same as when it was quite straight, and he quotes the case
of a pipe in which gentle bends of several degrees had no effect.
The fact that Rj^ is caused largely in the downstream tangent
(chap. ii. art. 13) was recognised in all the experiments, and it
was included in the observations, the normal loss of head due to
the tangent length being afterwards deducted. Brightmore found
that the loss of head in the bend itself was little, if at all, greater
than in an equal length of straight pipe, but the circumstances
seem to have been peculiar, as noted below.
In a cross-section a few feet downstream of the termination of
a 90-degree curve of 40-feet radius in a 30-inch pipe the maximum
velocity was found with low velocities to be in the centre of the
pipe, but it moved, when the maximum velocity was 3-5 feet per
second, to a distance from the edge of the pipe equal to about ^20
of the diameter. A further increase of 30 per cent, in the velocity
failed to shift it further. With curves of 15 feet and 40 feet
radius its position was about the same. In Brightmore's experi-
ments on 3-inoh and 4-inch pipes the flow iu a bend approximated
to that in a free vortex, i.e. the velocity in going across the pipe,
at the lower end of the bend, from the outside to the inside of the
bend, was nearly inversely as the radius struck from the centre of
the bend. He also found, with the 3-inch pipe, with R equal to
12D, that when /^exceeded 3 feet per second the condition was
unstable, Hj^ being sometimes about a mean between the values
for R=10D and li= 14D, but being sometimes much less.
Weisbach, as well as most of the other experimenters, make Hg
equal io Zi — . The following are the approximate values found
for Zi for bends of 90°: —
Experimenter.
Weisbach .
Davis . .
Brightmore
Do.
(Schoder
Williams '\
Hubbell [
and I
Fenkell J
Diameter
of pipe
(.U).
IncheH.
I)
6
i''i'nt.
1
Radius of Bend (ii).
Zero
(elbtiw).
2-bD.
1
S-&B. : bD.
■iD. \ 10Z>.
\iU.
VjD,
■1
■20D.
•98
•14
•135
•33
•i:.
•49 '..'.
::.
1-17
■20
•39 ...
■15
■1-J
•11
■35
■40
•14 ' -m
... i
■025
■015
•14
Whatever is known regarding the relative losses of head in
bends subtending dilferent angles is given in chan. ii. art,- i ^
PIPES
135
5. Relative Velocities in Cross-Section. — The velocities at
different points in the cross-section of a pipe have been observed
chiefly by means of the Pitot tube (chap viii. art. 14). Bazin found
that the velocity curve was convex downstream, and that r= -liR,
r being the distance from the axis to the point where the velocity
is equal to F — the mean velocity in the pipe — and B being the
radius of the pipe. In a 30-inoh pipe the form of the velocity curve
was found by Williams, Hubbell, and Fenkell to be very nearly
a semi-ellipse. The velocity ratios tended to become irregular
with low velocities. It is useless to discuss the precise nature
of the curve until the ratio of V to the central velocity is better
determined.
Regarding this ratio various old experiments show somewhat
conflicting results. The ratio increases with V and also with the
diameter of the pipe. The following table must be taken as show-
ing probable and approximate values only : —
Mean Velocities in Feet per Second.
Kind of Pipe.
of Pipe
in inches.
■78
1-6
2-S
8-6
5
s
14
62^5
Brass,
H
■84
Brass seamless, .
2
•70
■73
•77
•79
•80
Cast-iron, .
7-5
■80
■81
■82
■83
•84
Cast-iron, .
9-5
•80
•81
•82''
•83
•84
■85
Cast-iron with
deposit, .
New iron coated
with coal-tar, .
9-5
ri2
\ 16
i, 30
•75
•81
•83
•82
•83
■81
•83
•83
•84
•82
•84
■84
■85
■82
■85
■83
■83
•85
■83
■85
Cement,
31'5
•85
•86
New iron coated
with coal-tar, .
*
42
...
•86
...
Bilton's figures for small pipes, mostly oast-iron (Proc. Victorian
Inst, of Engineers, 1909, and Min. Proc. hist. G.E., vol. clxxx.), are
as follows : — ■
Central velocity, ft. per second,
J-inch pipe
|-inch pipe
1-inch pipe
IJ-inch pipe
2-inch pipe
3-inch-pipe
4 -inch pipe and larger
2.
4.
6.
8 and over
750
•764
•788
•804
780
•793
•817
•830
793
•810
■835
•848
807
■830
■855
■868
812
■839
•865
•878
•843
•872
■888
IT'
*»■•
•873
•890
136 HYDRAULICS
Bilton considers that the ratio diminishes slightly as the rough-
ness increases. In large pipes it was found that in two cases the
ratio was as much as 0-914 and 0-994. Bilton explains this by
suggesting that in large pipes the maximum velocity may not
always be at the centre of the pipe, but that, owing to obstructions,
oscillation may take place, and it may follow a wave-like course ;
in large pipes at low velocities the ratio is not definitely ascer-
tainable.
Section II. — Variable Flow
6. Abrupt Changes. — The losses of head occurring at abrupt
changes in small pipes have been found experimentally by
Weisbach, and are as below.
Abrupt Enlargement (Fig. 4, p. 5). — The loss of head is
(V —VY
"~ <i — O'' ^^^ head due to the relative velocity, but see remarks
in chap. ii. art. IS.'^
Abrupt Contraction (Fig. 3, p. 5). — The loss of head (and also for
a diaphragm (Fig. 90) or for a contraction with a diaphragm) is
chiefly caused by the enlargement from UF to MX, and is to be
found as above. To find the velocity at EF divide the velocity
-' c
--.Zl
PiQ. 90.
at MNhy c„. For a diaphragm" (Fig. 90) the values -of t\ were
found to bo as follows : —
Area 7^^.^ .2 .3 .4 .5 .g .^ .^ .g ^.q
Area C'V
r„=-624 -6.T2 -643 -Cno -681 -Tli! -755 -813 -89l' 1-00
These may bo accepted for the other cases.
Elbow (B'ig. 91). — The loss of head i.s
«„^ where .:;„=-9.|G sin"l-f 2-05 sin-^
2g -2 9.
' Seo iilso Notos at. cml of ohapti-r.
" See also ohaji. viii. art. 17.
PIPES
137
The values of s^ are as follows : —
6= 20° 40° 60° 80° 90° 100° 110° 120° 130° 140°
/„=-046 -139 -364 -740 -984 1-260 1-556 1-861 2-158 2-431.
Fig. 91.
Thus at a right-angled elbo-vv nearly the whole head due to the
velocity is lost. When two right-angled elbows closely succeed
each other the loss of head is double that in one elbow if the two
bends are in opposite directions, but is no greater than that in a
single elbow if the bends are both in one direction.
Gate-Valve (Fig. 92).—
^>«
7
8
.3
4
"=10
-948
-856
.f„ = -0
-07
-26
•740
•609
-466
-315
■159
-81 2-06 5-52 17-0 97-8.
Where A is the sectional area of the pipe and a that of the
opening.
1
I
X..
Fia. 92.
Fig. 93.
CocJc (Fig. 93).-
^ = 5° 10°
15°
20°
25°
30°
35°
40°
45°
.^= -926 -850
■772
-692
■613
-525
-458
-385
•315
z,= -05 -29
•75
1-56
3^10
5-47
9-68
17-3
31-2
c/> = 50° 55°
60°
65°
82°
±= -250 -190
-1.37
-091
00
x= 52-6 106 206 486
138 HYDRAULICS
Throttle Valve (Fig. 94).—
.^\
mmmm^mmmmimM.
--"-^-"--
mm^mmimmmmmiiiiM
Fio. 94.
</.= 5°
^(=•24
.^=55°
aj = 58-8
In the last
the loss of head.
It is not at all certain that the above figures apply correctly to
large pipes, and in fact it has been proved that some of them do
not apply correctly. For a gate in a 2-foot pipe z„ has been found
to be as below.
10° 15°
20°
25-
30°
35°
40°
45°
50°
•52 -90
1-54
2-51
3-91
6-22
10-8
18-7
32-6
60°
65°
70°
118
256
751.
ihree cases ^^ ^a
and Zi
are multipl
ed by
/-to
2^
give
D
as observed.
hy Weisbach's rule
given above.
XS
41-2
43
IJ.
31-35
28
I
22-7
IT
1
»
11-9
7-02
§
8-63
5-52
5/12
6-33
3-77
11/24
4-58
•J -87
1/2
3-27
206
7/12
1-55
111
2/3
•77
•57
10
•00
•00
When loss of head due to any of the above causes occurs, the
line of hydraulic gradient slinws a sudden drop as at GH, Fig. 95,
its inclination is reduced, and with it the velocity and discharge of
the pipe. If the local loss of head did not exist the slope would
be KL. The velocity to be used in calculating the loss of head is
that due to KG and not KL. If a second cause operates at M the
gradient becomes KG', Jl'-tll, NL, and the loss of head G'H' is now
less than before because the velocity is less. Thus the loss of
PIPES
139
Pig. 96.
head does not increase in proportion to the number of causes
operating. But where economy of head is desired, it is necessary to
avoid abrupt changes jr
of all kinds, using
tapering ' reducers '
where the diameter
changes, and curves
of fair radius at all
bifurcations or changes
in direction.
It appears that the disturbance of the velocity ratios due to
abrupt changes may extend downstream for long distances. Bazin
found that the disturbance from the entrance contraction of a
32-inch pipe disappeared at 25 to 50 diameters downstream, but
disturbance due to curves has been found to extend to 100 dia-
meters. In the disturbed region the pressures, as indicated by
pressure columns, appear to be below normal, or at least to be un-
reliable. In some important experiments on a 6-foot pipe ^ some
of the results are doubtful and probably erroneous, owing to a
piezometer being placed just downstream of an abrupt change.
7. Gradual Olianges. — When a gradual change occurs in the
sectional area of a pipe equation 16, page 22, must be used. At
a point where the diameter of a pipe changes a tapering piece is
usually put in. If the taper is gradual the loss of head in it from
resistances is about the same as in a uniform pipe with the same
mean velocity.
The following are examples of accidental changes in the dia-
meters of pipes : —
(1)
(2)
(3-4)
(5-6)
(7)
(8)
(9)
(10)
(11)
(12)
O
a
.2
p
■a
l=!
S
o
Ins.
Actual
Diameters.
Velocities.
or
Loss of
Head
from
Resist-
ances
or li' or
v^L
c-'Ji'
Actual
Fall in
Gradient
or h.
Percent-
age of
figure in
column
7 to
figure in
column
10.
V^
Ins.
<h
Feet.
^2
Feet.
Ins.
Feet.
Feet.
Feet,
Feet.
Feet.
100
12
12-5
11-5
4-0
4-73
-•099
4-38
113
■609
•708
10 2
25
30
29^ 301
4-0
3-93
+ -0082
3^97
128
•0384
•0302
2^4
25
30
29| 30J
1-0
•983
-f-OOOSl
•993
113
•00312
•00261
163
Transactions of the American Society of Civil Engineers, vol. xxvl,
140
HYDRAULICS
The figures in column 1 1 are obtained from those in columns 7 and
10. If the flow were uniform the figures in columns 10 and 11
would be the same, and the ratio of these figures to one another
shows the error caused by assuming the pipe to be uniform. If
the fall h is observed, and V found from A and G, the value of V
found will be erroneous in the ratio (neglecting the small variation
in C) of sjh to Jli!, that is, in the first of the cases shown, by
about 8 per cent, of the smaller figure. If h and V are observed
{V being found, say, by measuring Q in a tank) and is deduced,
the error in C will be similar to the above. If A is not observed,
but deduced from known values of V and 6', then the percentage
error is as shown in column 1 2. The second and third cases show
the same pipe with very different velocities, and it will be noticed
that the percentage of error does not vary very greatly. In the
first case quoted the variation of the diameter from the nominal
diameter is perhaps excessive and hardly likely to occur in practice.
With longer lengths of pipe the percentage of error will, of course,
be small, but sometimes observations are made on short lengths,
and it is clear that in such cases great error may arise, if the
diameter is assumed to be uniform.
When the diameter of a pipe is reduced (Fig. 96) the velocity
head in the narrow part is increased and the pressure head
;• ;
1
1 1
^__i._
1
1
Fig. 96.
reduced. The insertion of a portion like ACE in a pipe causes
vciy little loss of licail if the tapers are gradual. The case is
similar to that of a compound tul)e (chap. iii. art. 17). If CD is
PIPES
141
small enough, the pressure there will fell below the atmospheric
pressure P„, and if holes are bored in the pipe at this section no
water will flow out, but air will enter. The pressures on the
conical surfaces AGDB and GDFE balance one another, and the
water has no more tendency to push the pipe forward than it has
in a uniform pipe.
With the arrangement shown in Fig. 97, the orifices being
made to correspond as exactly as possible, the water flows with
very little waste into the second reservoir, and the head Gil is
slightly less than KL.
The pressure in the
jet KG is Pc,, and it
makes no practical dif-
ference whether this
portion is enclosed by
a pipe or not, so long
as the head KL is kept Fig. gr.
the same.
If at GD (Fig. 96) another pipe is introduced, pumping can be
effected through it, as with the case of a cylindrical or compound
tube.
When the hydraulic gradient of a pipe is so flat that the fall
between two pressure columns would be too small to be properly
observed, the 'Venturi Meter' (Fig. 96) is adopted. It consists of
two tapering lengths of pipe with two pressure columns. If
the diameters, velocities, and sectional areas at AB and GB are
D, V, A and d, V, a, then (chap, ii.)
.1g^ 2,
-+H.
Also
Therefore
(I
2g a" '2g
2g\a' )
A'-d-
a
(H-h).
j2g{II-h).
To allow for loss of head in the tube a co-efficient c must be
used, and
142 HYDRAULICS
If the pressure at CD is less than Pa, the height A^ measures the
diflference (the pressure tube being bent as shown by the dotted
lines), and h^ must be deducted from h^ to give h.
The length ^C is actually made less than CE. For other details
concerning Venturi meters see chap viii. art. 16.
Section III. — Co-efficients and Formula
8. General Information. — Pipes of importance are generally of
iron. Of these the vast majority are of cast iron. In America
some pipes — generally large — are of riveted steel or wrought iron,
aud some are wood-stave pipes. Pipes are also made of concrete or
are lined with cement. An iron or steel pipe if not protected by
an inside coating of asphalt — this term also includes coal tar and
other compositions — generally becomes affected in time by 'in-
crustation.' Even if so protected it often becomes affected by
incrustation or sometimes by vegetable growths. A ' clean ' pipe
is one — whether coated or unooated — not affected in any way or
which, if affected, has been cleaned. It is only for clean pipes
that definite co-efficients can be given. Others will be referred to
below (art. 10).
The sizes of pipes constantly tend to increase. There are
cast-iron pipes 5 feet in diameter. A concrete pipe 14'5 feet in
diameter is in use, also an 11-foot riveted steel pipe, lined with
concrete.
For each class of pipe there is a separate set of co-efficients. C
increases with R and also to some extent with S, that is with V.
In tables it is usual to show C for different values of V, not of .S".
There are few observations for high velocities. Oi-dinaiy velocities
range from 1 or 2 to 5 feet per second. Velocities of more than 10
feet per second are rare. Experiments on pipes have included
many sizes and many velocities. Very frequently there are several
values of S and V for one pipe. To obtain complete and accurate
sets of co-efficients reliable experiments should be made with a large
range of velocity on each one of a considerable number of sizes of
pipes. It cannot be said that this has been done. To a great
extent inference has to be adopted. Knowledge has, however, been
improved of late.
It has been shown above that a slight difference in D, or irregu-
larity in D, has a groat effect. It must be added that — at least as
regards some of the older experiments — the diameter may have
piPKS 143
been inaccurately stated, the manufacturer's size having been
accepted. There may be considerable difficulty in obtaining V or
Q with accuracy (chap. viii. section i.). Errors in the measurement
of Q, D, and S may be in either direction, those in S and J) being
relatively greatest with low values of these quantities. But error
may arise from unsuspected or unreported incrustation, air lock,^
losses of head from bends or obstruction by objects which have
accidentally got into the pipe or — in small pipes which cannot be
got at from inside — by projecting pieces of lead used for the joints.
All these tend to give low values of C. Hence, generally, G as
reported is likely to.be too low rather than too high, and to be
worst determined when S or D is small. Small channels are no
doubt more sensitive than large channels to variations in the
roughness.
From the point of view of economy it is important to obtain
reliable co-efficients for pipes. It is sometimes said that certain
co-efficients are 'on the safe side,' and sometimes a distinction is
drawn between ' laboratory ' and ' field ' experiments, the former
being those in which sources of error are carefully removed. The
value of C which is sought is the value for a clean pipe free from
sources of error. The engineer can make allowances, and can be
on the safe side as much as he thinks necessary.^ It is not right
to compel him to be so by supplying him with low figures.
Neither should he be supplied with too high figures. There will
always be a small margin within which co-efficients will vary. The
value sought is not the one at the highest edge of the margin. It
is one which will be obtained under proper conditions, and may
possibly be exceeded.
9. Co-efficients for Ordinary Clean Pipes. — For cast-iron pipes
Darcy obtained a set of co-efficients which vary from 93 to 113 as
-ffi varies from '042 foot to 1 foot. Smith and Fanning framed
much more extensive sets, making G increase with both Ji and V.
Their co-efficients apply to clean cast-iron, steel, or wrought-iron
pipes (not riveted), coated or uncoated, and with joints smooth and
curves of fair radius. Lawford framed a similar set of co-efficients.
Kutter's co-efficients (iV='011) are also much used. A brief ab-
stract of most of the above — for a velocity of 3 feet per second,
^ See notes at end of chapter.
^ In America a factor of safety — having reference to discharge and not to
strength — is in many cases adopted. Its value can be fixed with reference to the
injury likely to result from overestimation of the discharge.
144
HYDRAULICS
which is about the most useful value — and of seven other sets of
co-efficients is given in the following table : —
Pii'E Co-efficients ( F'= 3 ft. per second).
Diam-
otev In
feet.
Kutter.
Smith.
Law-
tord.
ria-
niant.
win.
Wil-
liams.
Saph and
Schoder.
Williams
and
Hazen.
Barnes.
Mai.
lett.
■25
1
4
10
40(?)
106
139
159
99
109
134
153
70
106
138
102
121
144
161
99
109
119
126
93
110
131
146
83
99
118
132
101
113
126
137
74
102
139
171
74
104
129
143
Fanning's co-efficients are nearly the same as Smith's. A set by
Tutton is nearly the same as Williams'. Smith's figures, obtained
by drawing a curve, included diameters up to 8 feet, but the curve
has been extended. In each of the seven sets mentioned, and in
Tutton's, G is obtained from a formula.^ It is not known that in
every case the author of the formula intended it to be applied to
the larger diameters included in the table. It is not known that
experiments have been made on any iron pipe, unless riveted, of
diameter greater than 6 feet, though experiments on larger circular
channels lined with mortar have been made, and some of the co-
efficients were meant to include such channels.
It will be seen that in some cases C is persistently high or low,
in others high or low for certain diameters. Some of the co-
efficients were clearly intended to be on the safe side or to allow for
badly laid or otherwise defective pipes. Unwin for large pipes
relied partly on an experiment which has been rejected by others.^
The small pipes on which Lawford experimented had been a year in
use. His co-efficients for such pipes — not for others — were rejected
by the author in 1911.^ They were, however, used by others, and
largely account for the low values of C, for small pipes, in the table.
Barnes gives new experiments by himself on 40-inch and 44-inch
pipes. His low figures for small pipes cause his curve to rim up
steeply, and give very high coefficients for a 10-foot pipe. Fig.
97a shows C for a few selected sets. The relative differences in
most cases are not very great, the zero being far below the diagram.
Kutter's co-eilicients ^ (C^-) were derived from observations on open
' The formula gives V, but C can be calculated from it For details and re-
ferences see art. 11.
" /iJm/iiieering, 2nd Juno 1911.
' Vav details as to Kutter's, Bazin's and Manning's co-efficients see chap vi.
arts. 11 to 13.
PIPES
145
channels of many sizes and degrees of roughness. It has long been
known to engineers that C'^-, supposing it to be correct for any large
smooth channel, is for the same kind of channel too low when B is
'\ 1 \ \
\\\ \
. ...Mu
\
^
\t t
= mm
;\
1
\\\v
\
,_::v^^i,
-->^
u 3 i i
8 3 s 3
about "25 foot or less. The left-hand part of the curve of C^ should
descend less abruptly. There is every probability of the true curve
being higher than most of the others. The curve now suggested
for acceptance is shown by a dotted line.^ For small diameters it is
near Smith's curve. For larger diameters it runs below the Kutter
1 Marked Smith-Kutter.
K
146
HYDRAULICS
line — but, as will be seen, C ,^ is in these cases somewhat high for
the particular velocity under consideration — and joins it when D is
13 feet.
Regarding the values of C for velocities other than 3 feet per
second, selected sets of co-efficients for various velocities are shown
c
V E L C 1 T 1 E S (ferf (jcr second)
I 234Se7e9K
160
iso'
140
150
120
110
100
asL.
SM«T'
.-J4U]
rTER.
,
.
1
4^
5^
.t-
.-'
/^■'
10-
FOOT
PIPE.
__,.--^
,^r-^
—
a-:^^
6<S!'
4^
'
--
^
,SJi-
^
^y^
t>
^
^
/ .
"- ff^
p^=
/
f
f
P^rifii!!
,'
/-
-— ^
>7
/
4-
-00 T
PIPE
/
■
,
-:-:::r
- —
..rtll
*L^--
■^
«iii>'
5
^!3J>
— ■ — '
- — "
''--
^
/
^
Kul
TER
■/I,*
>/
/
l-F
HOT F
IPE.
/
Fig. 97b.
in Fig. 97b for three sizes of pipe. The ordinates for velocities of 3
feet per second agree with those of Fig. 97a. ilallett's formula does
not provide for any alteration of C as V changes. It will be men-
tioned again (art. 11). In the other seven formulte C increases on
the average by 17 por cent, as F varies from 1 to 10 feet per second.
The increase is independent of D. Smith and Fanning have about
the same average rate of increase, but it is less as D is greater.
This is doubtless correct in principle, because in an open channel
PIPES 147
C ceases to increase when R is great. Knttcr makes it cease to
increase when E is 3 '28 feet, i.e. when D is, say, 13 feet. Accepting
this and considering all the figures, the co-efficients of table xxva.
are arrived at. The figures for very high and very low velocities
are of course not so well determined as the others.
The law of variation of Ck is peculiar and can hardly be correct.
It changes rapidly when V is low, and ceases to change when V is
higher. When 2> is 1 foot the Kutter curve is too low, as explained
above. For larger diameters up to 8 feet the agreement is very
much as when Z* is 4 feet. Owing to the bulge in the curve, C k is
relatively high when F is 3 feet per second. For high velocities
Ck is too low except when D approaches 13 feet.
Manning's adaptation of Ok does not vary with V. When V is
3 feet per second it agrees closely with Smith's co-efficients.
With regard to small pipes, Schoder and Gehring,'- with pipes —
mostly rusty — of diameters of 3 to 8 inches, found Fanning's figures
to be generally some 3 per cent, too low. They have been slightly
raised, except for the smallest sizes — the increase, when F= 3, is 5
per cent, for the 1-foot pipe, and 3 per cent, for the 6-inoh — and
this brings them into accord with those of the Smith-Kutter curve
for larger pipes. They are included in table xxva. Kutter's co-
efficients — corresponding to values of F, not of S — are given in
table xxvB. For Fanning's and Smith's original co-efficients see
tables xxiv. and xxv.
All the co-efficients apply to cast-iron, wrought-iron, or steel
pipes (not riveted), coated or uncoated, well laid, and with joints
smooth and curves of fair radius. I'hey apply to pipes of other
materials if N is -Oil. Kutter's co-efficients apply to all such
channels, with the reservations already made.
As regards any possible difference between a coated and an un-
coated pipe Smith, with a 1 'OS-inch pipe, found that coating it made
no difference. This was confirmed by the experiments of Schoder
and Gehring above referred to — some of the pipes were coated and
some uncoated — and it is confirmed by general experience. Most
of the largest pipes are coated, and experiments on such pipes when
uncoated are wanting.
Kutter's and the other co-efficients dealt with in chap. vi. were
meant to apply to open channels. Knowledge regarding small
open channels is derived chiefly from Bazin's experiments. In
these there are only a few cases in which, in the same channel, F
changes while R does not change. In only some of such cases is
^ Engineering Eecord, 29tli August 1908.
148 HYDRAULICS
there indication of increase of G with V. Kutter considered all
Bazin's experiments and others, and concluded that C increases
with V until S la \ in 1000. He clearly did not discover the exact
law. Experiments on pipes have been far more numerous, and
there are frequently, as has been seen, several values of S and V for
the same pipe, and thus clear evidence is obtained of the increase
of C with V, co-efficients such as those above discussed can be
obtained and the — not very great — inaccuracies of Kutter's co-
efficients corrected.
Let V be the velocity in a circular channel running half full. It
is improbable that V will be appreciably different^ — <S^ being the
same — when the channel is full. The distribution of the velocities
(chap. vi. section iii.) is not the same, but this can hardly affect
appreciably the general forward movement. The co-efficients of
table xxvA. are probably better suited than any others to open
channels of small or moderate size when iVis 'Oil.
For pipes of cement, mortar, concrete, or brickwork there are
Fig. 97c.
few experiments from which tables such as xxva. could be framed,
and Kutter's co-efficients should be used. They are given in table
xxvB. High velocities in such channels are unusual. For a given
material, e.g. brickwork, the degree of roughness is uot exactly the
same in all cases. For the selection of the proper value of N for
any pipe or channel see chap. vi. art. 12.
10. Co-efflcients for Other Pipes. — Riveted pipes are made up of
iron or steel sheets. The pipes are generally of large size, say,
2 to 10 feet in diameter. The sheets have lap joints longitudinally.
In the 'taper' pattern each length of pipe tapers slightly, the
smaller end fitting into the larger end of the next length down-
stream — as in a stove pipe — and being riveted to it. There is
thus a succession of abrupt but slight enlargements. In the
' cylindrical ' pattern each alternate length is made of larger
diameter so that the ends of both adjoining lengths fit into it, and
are riveted to it. There is thus a succession of enlargements and
contractions. In some pipes, however, there are butt joints.
There is also a 'locking bar' type of pipe (Fig. 9Tc) in which
the sheets, instead of being riveted longitudinally, are held in the
PIPES
149
grooves of a longitudinal bar. Usually there are double rows of
rivets, both longitudinally and at the joints. The larger the pipe,
the thicker generally the plates and the larger the rivet heads.
Thus the larger the pipe, the greater its general roughness is likely
to be.
The values of C as ascertained for riveted pipes of diameters
from 2'75 feet to 8-615 feet are erratic.^ Generally the change in
C with change of B is comparatively small. Sometimes the larger
diameter has the smaller value of C. All this is probably due to
the larger pipe being the rougher, and to the different patterns.
Whether the taper or the cylinder pattern gives the higher co-
efficient is not known. By taking values of C for all the diameters
c
Velocities (feert»r s»con<j)
1 2 3 4 5 «
110
no
loo
,
—
-J
"
..-^
^
y
4-FOOT PIPE (riveted")
r
Fig. 97d.
within the range mentioned above — the mean diameter is 4 feet —
and striking a general mean, a curve (Fig. 97d) has been arrived
at. The curve is flatter than the corresponding curve in Fig. 97b,
i.e. C. is less affected by changes in V. When F is 3 feet per
second the discharge of the riveted pipe is 20 per cent, less than
that of the cast-iron pipe. N is between 'OlS and -014. The
CO- efficient for a riveted pipe of any of the sizes above considered,
for any given value of F, will probably differ by not more than
5 to 7 per cent, from the corresponding figure on the diagram, but
it may be either more or less. Of the largest sizes one is more and
one less. And similarly with the smallest sizes. In designing a
riveted pipe, figures should be obtained for actual pipes of similar
pattern and size. Otherwise — and to some extent in any case —
the factor of safety should be higher than for a cast-iron pipe. The
' lis Experiments on Riveted Steel Pipes. Hydraulic Flow Reviewed.
Am. Soc. O.E., vols, xl., xliv., and others,
Trans.
150
HYDRAULICS
following table, obtained by calculation from Garrett's Hydraulic
Tables and Diagrams, is, however, given : —
l)i<iincl,ir of
1.
Velocity in Feet per Second,
3.
6.
8.
10.
i'm\.
1
102
104
14
94
105
107
2
96
106
108
112
2i
97
107
111
114
116
3
99
108
112
116
119
4
10.3
110 114
116
5
104
112 114
6
108
115 116
For smaller pipes sheet iron is used, and there may be single
rows of rivets. For such pipes, asphalted and with diameters of
10| inches to 25 J inches, and with V ranging up to 10, 12, and
20 feet per second. Smith found G to be very much the same as
for ordinary cast-iron pipes (table xxva.). The thickness of the
sheets was usually only -0054 foot to 0091 foot.
If a large riveted pipe is lined with cement so as to be made
uniform and smooth, the value of will be increased accordingly.
The discharge — allowance being made for the thickness of the
lining — is likely to be increased by some 20 per cent.
For small spiral riveted pipes C has been found by Schoder and
TABLE XXIlA.
Velocities in Feet per
1
Description of Pipe.
Joints.
Diameter
of Pipe.
Second.
Remarks.
1.
3.
5.
10.
Indies.
Spiral riveted,
Riveted
6
+ 3
+ 7
+ 9
+ 11
Steel 0-05 in.
asphalted.
flange.
thick.
Spiral riveted,
Do.
6
-5
- 6
- 6
9
Steel 0-078 in.
galvanised.
thick.*
Spiral riveted,
Do.
4
+
+ 3
+ 4
+ 5
Steel 0-0375
asphalted (flow
in. thick.
with the laps).
Spiral riveted,
Do.
4
-4
- 1
+
+ 2
Steel 0-0375
asphalted (ilow
in. thick. 1
against the
laps).
Seamless drawn
Special
5
+ 6
+ 13
+ 18
+ 23
Flange ar-
brass.
flange.
ranged so 1
as to give <
a continu- i
ous smooth
pipe.
Gehring to bo as given above '^ in table xxiiA. The figures show
the differences l)(jt\\i;on the e.xperimental co-efficients and those of
' Jiiiijiiieering llirvnl, 29tli August 1908.
PIPES 151
Fanning. In the case of the 6-inch pipes C was the same, whether
the flow was with or against the lap. The rivets had very flat
heads. ' The asphalt coating tends to fill up and smooth the lap,
but the galvanising leaves the edge of the lap sharp.'
For wood-stave pipes the results of a great number of experiments
are given by Scobey.^ The diameters ranged from 4 inches to
13 '5 feet. The values of G are in many cases extremely erratic.
Some of the observations were carried out under great climatic and
other difficulties. Sometimes the increase of C with V is very
rapid, but sometimes it is nil, and on the average it is about the
same as with cast-iron pipes, and the value of G for wood-stave
pipes should be taken as being 9 or 9-5 per cent, less than the
value shown in table xxva. In the Bulletin it is suggested that
the percentage averages about 4'5 when V= 3, about 1 when V= 7,
and about 7 when Y= 1, but this refers to discharges of cast-iron
pipes calculated by the Williams-Hazen formula. It will be seen
(Fig. 97a) that this agrees — owing to the shape of the Williams-
Hazen curve — with the figures now proposed. N for wood-stave
pipes is about •012. The discharging capacity of a wood-stave
pipe does not usually either increase or decrease with use. The
uncertainty as to G makes a comparatively high factor of safety
desirable in designing.
The deposits and growths in pipes, already referred to (art. 8),
are of various kinds and depend on the character of the water.
The reduction of discharge which they are likely to cause is a
matter of experience and judgment. Frequently there is a slimy
deposit. This may form on the inside of the coating of a pipe or
on iron, cement, or masonry. In time it may seriously reduce the
discharge. With some waters the slime is succeeded by nodules.
In some climates and with some waters vegetable growths occur
inside the pipe. They can be prevented by sterilising the water.
Incrustation of iron pipes is worst with soft moorland waters. If
there is no coating, or at small holes or cracks in the coating,
tubercles or nodules are formed. The nodules may be preceded by
slime. Limestone water is far less harmful and no coating may
be needed. In course of time the discharge of a tuberculated iron
pipe may be reduced by 30 per cent, or even, especially with small
pipes, by 50 per cent.
In an iron pipe slimy deposit may reduce N to about '013 —
that is, by some 16 per cent. — in a few years. On masonry and
cement it has less effect, perhaps because the channels are larger.
'■ U.S. Department of Agriculture, Bulletin No. S16.
152 HYDEAULICS
Brickwork may deteriorate with age independently of deposits
(chap. vi. art. 11). Barnes has f ound '^ that with the soft water
from Tliirlraere, in 40-inch and 44-inch asphalted mains, the
discharge was reduced by 13 per cent, in one year and by a smaller
percentage year by year, the total reduction in ten years being
31 per cent.
In America it is sometimes estimated that the discharge of a
cast-iron pipe is reduced by 15 per cent, in ten years and by
30 per cent, in twenty years, and that of a riveted steel pipe by
9 per cent, in ten years. ^
For a 2-inch seamless brass pipe Saph and Schoder found C to
exceed Fanning's figures, the excess being 18 per cent, when
V= 5"77 feet per second. See also table xxiiA.
Schoder and Gehring found that a 6-inch wrought-iron pipe in
long service in a steam-heating main had a sort of glaze inside it,
and C was some 16 per cent, higher than Fanning's figures.
For small tin, lead, zinc, or glass pipes Fanning's co-efficients are
fairly correct. For 2-5-inch hose they are nearly correct when the
hose is of rubber or lined with rubber, but they should be reduced
by about 16 per cent, when the hose is of linen and unlined.
11. Formulee. — The ordinary formula for flow in pipes is sometimes
put in the form — = ^^^s^- This gives the loss of head, IF, in a
given length when C and -ffi are known. If the diametere of two
pipes are equal, the loss of head is as -— ;^. A moderate difference
in estimating C — as when there is a choice of formulse — makes a
large diflference in II.
The formulae referred to in art. 9 are mostly of the form
S = Xj^, where m and n are quantities such as 125 and 1'85, and
K is constant. They are sometimes called exponential formulae.
There are formulae of this type for open channels and weirs. It is
unlikely that they are the true theoretical formulae. The main
idea is to avoid variable co-efficients. From the practical point of
view there are serious objections to the use of such formulae.
Instead of referring to tables of co-efficients it is necessary to use
a table of logarithms. The practical engineer has constantly to
iriake roii<;h and rapid calculations in connection, say, with changes
whicli arc contemplated or which have come about of themselves,
' Mill. Prof, fii.tl. C. E. , vol. coviii.
2 U.S. Diij)artraeiit of Agriculture, Bulletin A'o. 376.
PIPES 153
e.g. a change in the width of a channel or of the depth of water in
it. Within the range of depth, etc., with which he is concerned
the co-efficient may be nearly constant, or, if not, he knows in
what direction it changes. The simplicity of the formula is of the
first importance. Even the detailed calculations made at the
desk are done more quickly with the simple formulae than with
logarithms.
Again there is the question of comparisons. With the existing
formulae it is easy to make a comparison between the discharges,
say, of two pipes, one of cast iron and one of riveted steel. With
the exponential formulae no comparisons can be made without
working out the discharges. The values of the indices of B and S
for two formulae or two classes of pipes are difierent. The com-
parisons made above (art. 9) were not possible until C in the
ordinary formula had been calculated. With the present formulae
the engineer can choose any value of c or C in which he believes.^
Lastly, there are great numbers of persons — for instance, the
irrigation subordinates in eastern countries — who can understand
ifl in the weir formula but not ff*, nor could they use logarithms.
The formulae referred to in art. 9 are as follows : —
Author — Flamant.3 Unwin.^ Williams. -i Saph and Williams and Barnes.7
Schoder.5 Hazen.^
K- -00036 -0004 -00038 '000469 -000368 -000436
71=1-75 1-85 1-87 1-87 1-852 1-891
m=l-25 1-127 1-25 1-25 1-167 1-454
Tutton's formula 8 is 7=\iO R-<^^ S'^^. Mallett's formula ^ is of
the Bazin type, and C= ^^ (where N is Kutter's N), and
does not vary with 7. For oast-iron, concrete, and locking-bar
pipes in best order, a=172, p=l, 8 = 30, and iV^=-011. For
slightly incrusted pipes, a = 162, ^8= 1, 8= 30, and N= -013. For
pipes in worse condition there are other figures. The formula is
of a general and inclusive type and is meant to give fair approxi-
mations under very diverse conditions.
' When utilising the tables in the present work any value of c or C can be used.
' Annates des Fonts ei Ghaussies, 1892. Water, Deo. 1913.
' Industries, 1886.
* Trans. Am. Soc. C.E., vol. li.
"^ Trans. Am. Soc. C.E., vol. li. For ' commercial pipes' n varies from 1-74
to 20. The figures in art. 9 were obta,ined by taking n as 1 '87.
» U.S. Department of Agriculture, Bulletin No. 376.
' Hydraulic Flow Reviewed. Water, 15th June 1916.
^ Journal of Association of Engineering Societies, vol. xxiii.
^ Min. Froc. Inst. G.E., vol. coviii.
154 HYDRAULICS
Notes to Chapter V.
Air in Pipes (art. 1, p. 122). — The quantity of air which water
can hold in solution is greater, the greater the pressure and the
lower the temperature. At points of low pressure there is a
tendency for air to be disengaged from the water. Air, however
introduced, impedes the flow of water and reduces the discharge,
the condition being known as 'air lock,' and most likely to occur
with low pressures at ' summits ' such as G, and with low velocities
because the air is not then so quickly carried along or absorbed.
At summits on important mains there may be automatic air valves.
These allow any accumulated air to escape, and they allow air to
enter the main when it is emptied for repairs and to escape when
it is refilled. When the line of gradient is not far above the pipe,
simple stand-pipes (Fig. 5, p. 9) may be used.
Fipes above Line of Hydraulic Gradient (art. 1, p. 122). — The
pipe UTRM (Fig. 97e) lies above the line of hydraulic gradient.
Such oases are not common. The heights of any pressure columns
in NRM axe less than 34 feet, and the pressures less than atmo-
spheric. Air may thus be disengaged from the water. At any
defective joints water will not escape but air will enter. If there
is a summit such as S above the gradient line, arrangements must
be made for periodically drawing off the air accumulated there.
For this purpose an air vessel is attached at the summit. At its
lower side is a cock. A, opening to the pipe, at its upper side a cock,
B, opening to the outer air. One or other of these must be closed.
Suppose B to be closed and the vessel full of air at the same pres-
sure as that in the pipe. To get rid of air A is closed and B opened.
Through B water is introduced and the vessel filled, and the air in
it expelled. B is now closed and A opened. The water finds its
way into the pipe, and if there is air in the pipe it is displaced and
enters the air vessel. By repeating the above operation the pipe
can be kept free of air and air lock prevented. Another method is
to attach an air exhausting pump and remove air from the air vessel
until water is drawn at the pump. A pipe with a summit above
the gradient line is often called a syphon. Pumping or suction are
necessary in order to first start the flow in it.
Let a summit L be above the line xy. It is sometimes incorrectly
said that flow is impossible, the idea being that water flows along
the pipe ?INL with such velocity as to consume in resistance all
the head available. If air is regularly drawn off at L flow will take
place, thn gradient being xL. Flow is impossible only when L is
higher than so.
PIPES
155
Abrupt Enlargement (art. 6). — Observations by Archer'- show
that the loss of head in pipes 1 to 3 inches in diameter was actually
(V - V )
B i^ — 1 ~ , where B was as follows : —
2<7
Ratio of
Ai to A^.
Values ol Vi, feet per second.
o_
(J.
12.
30.
80.
1 : li
1 : 4
1 : 00
1-225
1-055
1-022
1-123
•965
■937
1-060
■911
■884
■981
•846
■820
•903
•780
•759
Fluid Friction (chap. ii. art 9). — The friction of water on a plane
surface is seldom exactly as V% but is as 1" where n varies from
about 1-7 to 2^16. See chap. s. art. 5, final paragraph.
Fio. 97e.
lllXAMPLES
Explanation. — The problem to be solved may be either to find
the discharge in a pipe for which all the data are known, or when
the discharge and one of the quantities Z> or S are known, to find
the other. In the first case the solution is direct, in the others
(since B and G vary with J) and S) indirect. The methods to be
adopted will be clear from the following examples.
In the examples Smith's and Fanning's co-efficients happen to
have been used, but of course the new Smith-Kutter co-efficients
— or any others — can be used in exactly the same manner.
1 Proc. Am. Soc. C.E., vol. xxxix.
156 HYDRAULICS
One advantage of the system of tables here adopted, as com-
pared to some others, is that V always enters as a factor. It is a
distinct advantage, in designing, that the value of V, and not only
of Q, should constantly come to notice.
Example 1. — Using Smith's co-efficients, find the discharge of a
C.-I. pipe whose diameter is 3 feet and slope 1 in 1000.
From table xxiv., G is about 123'5 and J^ about 3 '4. Smith's
co-eflficient for this value of F^is 130, so that ?^will be about 3'6
and C about 130. From table xxiii. ^^=-866. From table xxvi.
(7^i?=112-5. From table xxviii. F=3-5&, which agrees nearly
with the value assumed, and confirms the co-efficient 130. From
table xxiii. J = 7-07. Then Q=7-07x3-56 = 25-17 c. ft. per
second.
Example 2. — Using Smith's co-efl&cients, design a pipe to carry
20 c. ft. per second, the fall being 10 ft. in 5000.
Assume D=2 ft. From table xxiii. A — 3-14:2 sq. ft. and
20
JR=-101. Also F=g7Yl=6'37 ft. per second. From table xxv.
C=129. From table xxvi. GJR=^\-2. This value does not
appear in table xxviii. ; .•. look out 182 '4, wHich gives (for S—-^^)
F=8'16; .•. V is 4'08, which is too low, that is, the assumed
diameter was too small.
Let i) = 2-5 ft. From table xxiii. ^=4-91 and ^/i?=-791.
20
Also J^=-- =4'07 ft. per second. From table xxv. C:=128.
From table xxvi. CJR=l(il. From table xxviii. r=4-52 ft.
per second, which is too high. The diameter 2o ft. is thus
slightly in excess of what is required. To find the actual dis-
charge, C (for F=4-5) is 129-5, CJR is 102-4, F is 4-58, and
(3 is 4-58x4-91 = 22-49 c. ft. per second.
/2'-4"\^ /14\^ l''-5
Since (^y^„) =\^) =~^r' "^^rly, .-. a 2 ft. 4 in. pipe -would
be too small.
Example 3. — A H-ft. C.-I. pipe has to carry a discharge of
18 c. ft. per second. What will the gradient be? Fanning's
co-efficient to be used. From table xxiii. ./ = r77. Then
V= ^^^ = 10-2 ft. per second. From table xxiv. C=117 and
1-77
iS'= -020 nearly. From table xxvii. ^<S'= -1414. From table xxiii.
pii'ES 157
7iJ=-612. From table xxvi.CVii=71-6 and 71-6 X -1414= 10'23.
Therefore S= -020 is correct.
Example 4. — A pipe 2 in. in diameter and 20 ft. long connects
two reservoirs, the head being 1 ft. and the pipe projecting into
the upper reservoir. Find the discharge, using Fanning's co-
efficients.
The pipe being short, the loss of head at entrance must be
allowed for. This (art. 1) is «„=1'93~-. Suppose /^to be 4 ft.
per second. Then from table i. - -=-25 and s„ is -48. This loss
occurs in the length of, say, '4 ft., so that i=19-6 ft. and
g=-^"^~"^^=-027. From table xxiv. ^=-040 is the slope which
19-6 ^
gives V=4:-0, so that Fhas been assumed too high.
Let V be 3'5 ft. per second. Then -^ = -19, and z. is '37, and
2g
S= = '032. Table xxiv. does not give this slope exactly,
but evidently C is about 97. From table xxiii. JE is '204. In
table xxvi. look out -408. Then CJE is ?|:^ = 19-8. The slope
(?= '032 is steeper than those in the tables. Therefore calculate
^S, which is -18, and CVi^ which is 19-8 X -18, or 3-56 ft. per
second, which is near enough.
Example 5. — An open stream discharging 16 c. ft. per second
is passed under a road through a syphon or tunnel of smooth
plastered brickwork of section 2 f t. x 2 ft., which first descends
10 ft. vertically, then travels 80 ft. horizontally, and again rises
10 ft. vertically, the bends being right-angled and sharp. What
is the loss of head in the tunnel ?
Here F'=-^^=4 ft. per second. There are 4 elbows of 90° each.
That at the entrance to the tunnel is opposite in direction to
the second. Hence the total loss of head from the elbows is
4X-984X— =-984ft.
To find the approximate loss of head from friction let Fanning's
co-eflicients be used. Then i^= -5, C= 1 17, S= -0024. The fall in
100 ft. is -24 ft. The total loss of head is thus ■98-|--24 = l-22 ft.
158
HYDRAULICS
Table XXIIL— Values ov .1
AND K
FOR Circular Pipes.
Diameter
Sur-tinnal
Area (.4).
Hydraulic
Radiuu («).
VU
Remarks.
Feet
Inches.
Square Feet.
Feet.
1
•00136
■0104
•102
1
■00307
' ^0156
■125
1
■00545
-020S
•144
H
■00852
-0200
•161
i|
•0123
•0312
•177
If
■0167
•0364
•191
2
•0218
•0417
•204
2^
•0341
-0521
•228
3
•0491
-0625
•250
4
•0873
-0833
•289
5
•136
-104
•323
Diameters not given in
6
•196
■125
•354
Table. To find A for a
7
■267
-146
•382
8
•349
-166
•408
larger diameter, look out i
9
•442
•187
•433
A for half the diameter 1
10
11
•545
■660
•208
-229
•456
•479
and multiply by 4. For
■785
•250
•50
a smaller diameter, look
1
■922
-271
•520
out A for double the
2
1^069
-292
•540
diameter and divide by
3
1^227
•313
•559
4
1^396
•333
•577
4. To find ^/R for a
. 5
1^576
•354
•595
larger diameterilook oat
6
7
V767
r969
•375
•396
•612
■629
^R for one-fourth the
8
2^181
■417
■646
diameter and multiply
9
2-405
•437
■662
by 2. For a smaller
2
10
2640
3^142
•458
•500
■677
■707
diameter, look out ,^/B
2
2
3^687
■542
•736
for 4 times the diameter
2
4
4^276
•583
•764
and divide by 2.
2
2
6
8
4^909
5^585
■625
•667
■791
•817
Circular Channels 7iot
2
10
6^305
•708
•841
full. For a channel of
3
7-069
•750
■866
circular section running
3
3
8 ■296
■812
•901
3
6
9^G?1
•S75
•935
half full, A is oue-half
3
9
11-05
•937
•967
of the value in the
4
12^57
10
1^0
table, and i^S is the
4
6
15^90
M25
i^oei
5
19-64
125
MIS
same as in the Table.
5
6
2376
1^375
1173
6
28-27
1^50
\'2'25
6
6
33-18
r625
1-275
7
38-48
1 ^75
1-323
7
C
4418
l'S75
1 •.■!70
8
50-26
2
1 414
8
(■)
56-74
2^1 25
1^458
9
03-62
2-2r>
1-5
9
6
70-88
2-375
1^541
10
78-54
2^50
1-581
PTPES
159
Tables XXIV. to XX Vb. — Co-Efficients for Pipes cokee-
SPONDING to given DiAMETEES AND VELOCITIES. (ArL. 9.)
(Also suitable for open channels when R is the same and N the saine.)
Tables xxiv. to xxva. are for ordinary pipes, N being about •Oil.
The small figures in table xxiv. show, nearly, the slopes which give the
velocities entered in tlie heading, and they can be used to show the
approximate slopes when the co-efficients in table xxv. or xxvA. are used.
XXIV. — Farming's Go-Efficients.
Velooit
ies in Feet per Second.
Dia-
meter
ofPipe.
■1
'■>
1
^
3
4
6
10
15
20
Inches.
4
43
51
76
87
93
94
96
100
102
103
■oier
049
098
•167
■371
93
i
50
75
79
88
93
96
98
101
103
104
•0007
032
066
•112
■243
62
I
73
77
81
89
94
94
98
102
104
105
■0083
024
048
•083
■180
47
H
77
81
86
90
016
94
033
96
■067
100
■120
102
32 ■
104
105
2
85
88
90
94
010
96
022
98
■040
101
■072
104
23
106
106
3
89
92
93
96
007
98
016
100
■027
102
■056
105
143
106
106
4
93
93
95
97
0049
100
oil
102
■019
103
■039
106
107
108
108
6
94
95
97
100
0032
102
0070
103
•012
106
■026
108
070
109
HI
8
96
97
99
102
104
105
107
110
112
113
Feet.
0024
0050
■0087
■019
060
1
93
100
102
105
0015
106
0032
108
■0064
110
■012
114
032
115
116
1-5
104
106
109
0092
HI
0019
113
■0084
114
■0074
117
020
118
...
2
109
111
114
00063
116
0014
117
■0024
118
■0051
121
014
122
3
117
118
121
00038
123
00082
124
•0014
127
■0030
128
0082
129
4
127
128
129
00024
131
00062
132
■00O94
135
■0020
135
0052
136
5
134
135
136
0OO17
137
0O040
137
00067
138
■0O14
142
0040
142
6
137
137
137
00014
140
00032
141
•00068
143
■0012
147
0031
147
...
7
141
143
143
00011
146
00026
147
■00045
148
■00097
151
0124
151
8
149
150
lol
00009
151
00020
152
•00034
155
158
158
...
160
HYDRAULICS
XXV. — Smith's Co-efficients.
Dla-
Velocities in Feet ptir Second.
meter
ofFipe.
1
2
3
4
6
6
8
10
12
15
20
Feet.
•05
78
82
86
88
89
91
91
91
91
■1
80
89
94
97
99
101
103
105
105
105
1
96
104
109
112
114
116
119
121
123
124
124
1-5
103
111
116
119
121
123
126
129
130
132
133
2
109
116
121
124
127
128
132
135
136
138
2-5
113
120
125
128
131
133
136
139
141
143
3
117
124
128
132
134
136
140
143
145
147
3-5
120
127
131
135
137
139
142
146
149
151
4
123
130
134
137
140
142
146
150
152
153
5
128
134
139
142
145
147
150
155
...
6
132
138
142
146
148
151
155
7
135
141
145
148
151
8
138
143
148
151
153
...
Notes on Hydeaulic Tables.
The tables in this book, .as already noted, admit of the use of
any co-efficient which may be selected. The examples given show
how they are to be used. ,
As regards interpolations, these can often be made by mere in-
spection. When strict accuracy is required the following example
{table xxvi.) may be followed. Let C be 109-7 and ^'^ be 1-118.
The upper and lower figures of C and GJB are taken from the
tables and the differences entered in the last line.
Diff.
CyJR
Diff.
109
121-8
■7
•8
109-7
122-6
-3
-3
110
12i-9
Total,
1-0
1-1
The 109-7 is interpolated, the differences entered in column 2,
tlie approximately proportionate differences in column 4, and the
figure 122-6 arrived at. To interpolate between two values of S
or JS (table xxviii.) proceed similarly, but if there is also an in-
terpolation in G it may be best to calculate V for both the values
oi sJS and then interpolate.
PIPES
161
XX Ta. — Smith-Kutter Co-efiicients.
Velocities in Feet per Second.
Diameter
of Pipe.
1
2
3
8
7
10
IB
20
Indies.
\
77
87
92
96
99
100
101
102
1
80
88
93
97
100
101
103
104
1
82
90
94
98
101
103
105
106
^
86
92
95
99
102
105
107
108
2
90
94
97
101
104
106
108
111
3
93
96
99
103
106
108
111
114
4
95
98
101
105
108
110
113
115
6
97
101
103
107
110
113
115
118
8
99
104
106
110
113
115
118
120
Feet.
1
102
107
111
115
118
120
123
125
1-5
107
113
116
121
124
127
130
133
2
113
119
122
126
129
132
135
2-5
118
124
127
131
134
137
140
3
122
127
131
135
138
141
144
3-5
125
131
134
138
142
144
4
128
134
137
142
145
148
5
133
139
143
147
150
153
6
138
143
147
152
155
158
7
143
147
151
155
158
161
8
147
151
154
158
160
150
153
156
159
161
10
153
156
158
160
162
11
157
158
160
161
163
12
160
161
162
163
164
13
164
164
164
164
164
162
HYDRAULICS
XX VB.—Kutter's Co-efficients.
Diameter
ot Pipe.
■612
Velocities in Feet per Secjnd.
1 2
3
i
1
2
3
i
1
2
3 : 4
1-5
138
147
009)
148
148
124
129
010)
131
131
(iV=-011)
111,116 116 116
2
•707
143
154
154
157
128
136
139
142
1171122 124 124
3
•866
150
163
166
168
135
145
149
149
121 i 130 132 132
4
1-0
157
169
173
174
141
152
155
155
1281137 139 140
5
1-118
164
173
178
179
147
157
160
160
135 1 142 144 144
6
1-225
170
177
182
184
153
161
163
164
140 ! 147
148 148
8
1-414
181
186
190
190
162
168
169
171
148 : 151
154 154
10
1-581
100
192
195
105
170
173
175
176
154 . 158
159 159
12
1-732
197
198
199
199
177
178
179
179
159 1 162
163 163
16
2-0
210
209
208
207
191
189
187
186
172 i 171
170 169
1-5
-612
100
104
•012)
104
104
93
(.V =
96
■013
100
100
1
2
-707
106
HI
112
112
97
101
106
106
3
-866
113
119
121
121
103
109
110
110
4
1-0
118
125
127
127
109
114 i 116
117
5
1-118
122
129
132
132
113
118 i 120
121
..
... 1 .
6
1-225
128
132
135
136
116
121 ' 123
124
8
1-414
136
138
140
141
122
128 129
129
'
10
1-581
142
144
145
146
129
132 , 133
133
12
1-732
148
148
149
149
136
137 138
138
16
2-0
160
157
156
155
147
145 144
143
".'. I ;
Note. — When f^ exceeds 4 feet per second G generally remains the same.
PIPES
163
Table XXVI. — Values of C JR tor various Values
op c and jr.
For a value of (7 lower than 90 look out double the value and halve the
result.
For a value of G over 140 look out half the value and double the result.^
Values of -j'R.
Values
of C.
•354!
•3S2
•408
■433
■456
•479
■500
•620
■640
90
31-!)
34-4
36'7
39-0
41-0
43-1
45-0
4rr8
486
91
■A-1-2
34-8
37-1
39-4
41-5
43-6
45-5
47 •S
49^1
92
32-6
35-1
37-5
39-8
42-0
44-1
46-0
47-8
49^7
93
32-9
35-5
37-9
40-3
42-4
44-5
46-5
48^4
50^2
94
33-3
35-9
3ji'4
40-7
42-9
45-0
47-0
48-9
50^8
95
33-6
36-3
38-8
41-1
43-3
45-5
47-5
49^4
51^3
96
34-0
36-7
39-2
41-6
43-8
46-0
48-0
499
51^8
97
34-3
37-1
39-6
42-0
44-2
46-5
48-5
50-4
52^4
98
34-7
37-4
40-0
42-4
44-7
46-9
49-0
51 •O
52^9
99
350
37-8
40-4
42-9
45-1
47-4
49-5
51-5
53-5
100
35-4
38-2
40-8
43-3
45-6
47-9
50-0
52^0
54^0
101
35-8
38-6
41-2
43-7
46-1
48-4
50-5
52-5
54^5
102
361
39-0
41-6
44-2
46-5
48-9
510
53-0
55-1
103
36-5
39-3
42-0
44-6
47-0
49-3
51-5
53-6
55^6
104
36-8
39-7
42-4
45-0
47-4
49-8
52-0
54-1
56^2
105
37-2
40-1
42-8
45-5
47-9
50-3
52-5
54-6
56^7
106
37-5
40-5
43-2
45-9
48-3
50-8
53-0
55^1
57^2
107
37-9
40-9
43-7
46-3
48-8
51-3
53-5
556
57-8
108
38-2
41-3
44-1
46-8
49-2
51-7
54-0
56-2
68^3
109
38-6
41-6
44-5
47-2
49-7
52-2
54-5
56-7
58-9
110
38-9
42-0
44-9
47-6
50-2
52-7
550
57-2
594
111
39-3
42-4
45-3
48-1
50-6
532
55-5
57-7
59-9
112
39-6
42-8
45-7
48-5
511
53-6
56-0
58-2
60-5
114
40-4
43-5
46-5
49-4
520
54-6
57-0
59-3
61-6
116
411
44-3
47-3
50-2
52-9
65-6
58-0
60-3
62-6
118
41-8
45-1
48-1
5I-1
53-8
56-5
59-0
6r4
63^7
120
42-5
45-8
49-0
52-0
54-7
57-5
60-0
62-4
64-8
122
43-2
46-6
49-8
52-8
55-6
58-4
61-0
63-4
65-9
124
43-9
47-4
50-6
53-7
56-5
59-4
62-0
64-5
670
126
44-6
48-1
51-4
54-6
57-5
60-4
63-0
65^5
68-0
128
453
48-9
52-2
55-4
58-4
01-3
64-0
666
69-1
130
46-0
49-7
530
56-3
59-3
62-3
65-0
67-6
70-2
132
46-7
50-4
53-9
57-2
60-2
63-2
66-0
68-6
71-3
134
47-4
51-2
54-7
58-0
61-1
64-2
670
69-7
72-4
136
48-1
52-0
55-5
58-9
620
651
08-0
70-7
734
138
48-9
52-7
56-3
59-8
62-9
661
69-0
71-8
74-5
140
49-6
53-5
57-1
60-6
63-8
67-1
70-0
72^8
75-6
^ Or look out J or f value and multiply accordingly.
^ For a lower value, e.g. '204 (see tabic xxiii.), lookout "408.
164
HYDRAULICS
Table XXVI. — Continued. — Values of C^B for various
Values of C and JE.
For a value ot G lower than 90 look out double the value and halve the
result.
For a value of G over 140 look out half the value and double the result.
Values of ^/n.
Values
otC.
•559
■577
■695
•612
•629
•646
■662
■677
•707
90
50-3
51-9
53-6
55-1
56-6
58-1
59-6
60-9
63-6
91
50-9
52-5
54-2
65-7
57-2
58-8
60-2
61-6
64-3
92
51-4
53-1
54-7
56-3
57-9
59-4
60-9
62-3
65-0
93
52-0
53-7
55-3
57-9
58-5
60-1
61-6
63
65-8
94
52-5
54-2
55-9
57-6
591
60-7
62-2
63-6
66-4
95
531
54-8
56-5
58-1
59-8
61-4
62-9
64-3
67-2
96
53-7
55-4
571
58-8
60-4
62-0
63-6
650
67-9
97
54-2
56-0
57-7
59-4
610
62-7
64-2
65-7
68-6
98
54-8
56-5
58-3
60-0
61-6
63-3
64-9
66-3
69-3
99
55-3
57-1
58-9
60-6
62-3
640
65-5
67
700
100
55-9
57-7
59-5
61-2
62-9
64-6
66-2
67-7
70-7
101
56-5
58-3
601
61-8
63-5
65-3
66-9
68-4
71-4
102
57-0
58-9
60-7
62-4
64-2
65-9
67-5
691
72-1
103
57-6
59-5
61-3
630
64-8
66-5
68-2
69-7
72-8
104
58-1
60-0
61-9
63-6
65-4
67-2
68-8
70-4
73-5
105
68-7
60-6
62-5
Gi-3
66-0
67-8
69-5
711
74-2
106
59-3
61-2
631
64-9
66-7
68-5
70-2
71-8
74-9
107
59-8
61-7
63-7
65-5
67-3
69-1
70-8
72-4
75-7
108
60-4
62-3
64-3
661
67-9
69-8
71-5
73-1
76-4
109
60-9
62-9
64-9
66-7
68-6
70-4
72-2
73-8
77-1
110
61-5
63-5
65-5
67-3
69-2
71-1
72-S
74-5
77-8
111
62-1
64-1
66-1
67-9
69-8
71-7
73-5
75-2
78-5
112
62-6
64-6
66-6
68-5
70-4
72-4
741
7.">-S
79-2
114
63-7
65-8
67-8
69-8
71-7
73-6
75-5
77-2
80-6
116
64-8
66-9
690
71-0
730
74-9
76-8
78-5
82-0
118
660
6S-1
70-2
72-2
74-2
76-2
78-1
79-9
83-4
120
67'1
69-2
71-4
73-4
75-5
77-5
79-4
81-2
84-8
122
68-2
70-4
72-6
74-7
76-7
78-8
80-8
82-6
86-3
124
69-3
71-5
73-8
75'9
78-0
80-1
82-1
83-9
87-7
126
70-4
72-7
75 '0
77-1
79-3
81-4
83-4
85-3
891
128
71-6
73-9
76-2
78-3
80-6
82-7
84-7
86-7
90-5
130
72-7
7r)'0
77-4
79-6
81-8
84-0
861
88-0
91-9
132
73-8
76-2
78-5
80-8
830
85-3
87-4
89-4
93-3
134
74-9
77-3
79'7
820
84-3
86'6
887
90-7
94-7
136
76-0
78'5
809
8312
85-5
87-9
90-0
921
96-2
138
77'1
70-6
82-1
84-5
86-8
891
91-4
93-4
97-6
140
78'3
80-8
83-3
85-7
88-1
90-4
92-7
94-8
990
PIPES
165
Table XXVI. — Continued. — Values of G JB fok various
Values of C and JR.
For a value of C lower than 100 look out double the value and halve the
result.
For a value of O over 160 look out half the value and double the result.
Values of V-R-
Values
ofC.
■736
•764
■791
■817
•S41
•866
-901
100
73-6
76-4
79-1
81-7
84-1
86-6
90-1
101
74-3
77-2
79-9
82-5
84-9
87-5
91-0
102
75-1
77-9
80-7
83-3
85-8
88-3
91-9
103
75-8
78-7
81-5
84-2
86-6
89-2
92-8
104
76-5
79-5
82-3
85-0
87-5
90-1
93-7
105
77-3
80-2
83-1
85-8
88-3
90-9
94-6
106
780
81-0
83-8
86-6
89-1
91-8
95-5
107
78-8
81-7
84-6
87-4
90-0
92-7
96-4
108
79-5
82-5
85-4
88-2
90-8
93-5
97-3
109
80-2
83-3
86-2
89-1
91-7
94-4
98-2
110
81-0
84-0
87-0
89-9
92-5
95-3
99-1
111
81-7
84-8
87-8
90^7
93-4
96-1
100-0
112
82-4
85-6
88-6
91-5
94-2
97-0
100-9
113
83-2
86-3
89-4
92-3
95-0
97-9
101-8
114
83-9
87-1
90-2
93-1
95-9
98-7
102-7
115
84-6
87-9
91-0
94-0
96-7
99-6
103-6
116
85-4
88-6
91-8
94-8
97-6
100-4
104-5
118
86-8
90-2
93-3
96-4
99-2
102-1
106-3
120
88-3
91-7
94-9
98-0
1009
103-9
108-1
122
89-8
93-2
96-5
99-7
102-6
105-6
109-9
124
91-3
94-7
98-1
101-3
104-2
107-3
111-7
126
92-7
96-3
99-6
102-9
105-9
109-0
113-5
128
94-2
97-8
101-2
104-5
107-6
110-8
115-3
130
95-7
99-3
102-8
106-2
109-3
112-5
117-1
132
97-2
100-8
104-4
107-8
111-0
114-2
118-9
134
98-6
102-4
106-0
109-4
112-7
115-9
120-7
136
100-0
103-9
107-5
nil
114-3
117-7
122-5
138
101-6
105-4
109-1
112-7
116-0
119-4
124-3
140
103-0
106-9
110-7
114-3
117-7
121-2
126-1
142
104-5 .
108-4
112-3
1160
119-4
122-9
127-9
144
105-9
110-0
H3-9
117-6
121-1
124-7
129-7
146
107-4
111-5
115-5
119-2
122-7
126-4
131-5
148
108-9
113-0
117-0
120-9
124-4
128-1
133-3
150
110-4
114-6
118-6
122-5
1261
129-8
135-1
152
111-8
1161
120-2
124-1
127-8
131-6
136-9
154
113-3
117-6
1218
125-7
129-4
133-3
138-7
156
114-8
119-1
123-3
127-4
131-1
1350
140-5
158
116-3
120-7
124-9
129-1
132 8
136-7
142-3
160
117-7
122-2
126-5
130-7
1 34-5
138-5
144-1
166
HYDKAULICS
Table XXVI. — Continued. — Values of CJB for various
Values of C and JIl.
For a value of G lower than 100 look out double the value and halve the
result.
For a value of G over 160 look out half the value and double the result.
Values of \/B.
Values
ofC.
•935
•967
1-00
1-061
i-iis
1-173
1-225'
100
93-5
96-7
100-0
100- 1
111-8
117-3
122-5
101
94-4
97-7
101-0
107-1
112-9
118-.1
123-7
102
95-4
98-6
102-0
108-2
114-0
119-6
124-9
103
96-3
99-6
103-0
109-3
115-1
120-8
1261
104
97-2
100-6
104-0
110-3
116-2
121-9
127-4
105
98-2
101-6
105-0
111-4
117-3
1231
128-6
106
99-1
102-6
106-0
112-4
118-5
124-3
129-8
107
100-1
103-5
107-0
113-5
119-6
125-'5
131-0
108
100-9
104-4
108-0
114-5
120-7
126-6
132-3
109
101-8
105-4
109-0
115-6
121-8
127-8
133-5
110
102-8
106-3
iio-o
116-7
122-9
129-0
134-7
111
103-7
107-3
111-0
117-8
124-0
130-2
135-9
112
104-7
108-3
112-0
118-8
125-1
131-3
137-1
113
105-6
109-2
113-0
119-9
126-2
132-5
138-3
114
106-5
110-2
114-0
120-9
127-3
133-6
139-6
115
107-5
111-2
115-0
122-0
128-4
134-8
140-8
116
108-4
112-1
116-0
123-0
129-6
136-0
142-0
118
110-3
114-0
118-0
125-1
131-8
138-3
144-4
120
112-2
116-0
120-0
127-3
134-1
140-7
1470
122
114-1
117-9
122-0
129-4
136-3
143-0
149-4
124
115-9
119-8
124-0
131-5
138-5
145-3
151-9
126
117-8
121-7
126-0
133-6
140-7
147-6
154-3
128
119-6
123-7
128-0
135-7
143-0
150-0
156-8
130
1-21-5
125-6
130-0
1,37-8
145-2
152-3
159-2
132
123-8
127-6
132-0
140-0
147-5
154-7
161-6
134
125-2
129-5
1340
142-1
149-7
1570
164-0
136
127-1
131-5
136^0
144-2
152-0
15!)-4
166-5
138
129-0
133-4
138-0
146-3
1.^4-2
161-7
168-9
140
130-9
135-3
140-0
148-5
156-5
164-2
171-5
142
132-8
137-2
142-0
150-6
158-7
166-5
173-9
144
134-6
139-1
144-0
15-2-7
160-9
168-8
170-4
146
136-5
111-0
i4!;-o
154-8
163-1
171-1
178-8
148
138-3
143-0
14S-0
156-9
165-4
173-5
181-3
150
1(0-2
144-9
150-0
159-0
lC.7-6
175-S
183-7
152
142-0
l<|-,-9
152-0
101-2
169-9
17S-2
186-1
154
143-9
148-8
r.4-0
163-3
172-1
180-5
188-5
1.56
145-8
150-8
150-0
165-4
174-4
182-9
191-0
1.58
147-7
152-7
15S-0
167-5
176-6
185-2
193-4
160
149-0
154-7
160-0
169-7
178-8
187-6
196-0
1 For a higher viilue, e.g. 1 -581 (see table xjiii.), look out -791.
PIPES
1G7
Table XXVII.— Values of S and ^S.
(For steep slopes not included in Table xxviii.)
To fiud ^S for a steeper slope, look out a slope 4 times as flat' and
multiply ^S by 2. Thus, for 1 in 50, s/S is -07071 x 2= -14142.
Slope
lin
Fall per Foot
orS.
VS
Slope
1 iji
Pall per Foot
orS.
v'S
100
-010
1
230
-004348
■06594
105
-0095238
09759.
240
-004167
•06455
110
•009091
095346
250
■004000
•06325
115
-008696
093250
260
■003847
•06202
120
-008333
091287
270
•003704
-06086
125
-008
089442
280
•003571
-05976
130
-007692'
08771
2P0
•003448
-05872
135
•007407
OS607
300
•003333
-05774
140
■007143
08452
310
•003226
-05680
145
-006897
08305
320
■003125
•05590
150
-006667
08165
33i)
•003030
•05505
155
•006452
08032
340
•002941
•05423
160
-00625
07906
350
•002857
-05345
165
•006061
07785
360
•06278
-05271
170
■005882
07670
370
•002703
•05199
175
■0057,14
07559
380
•002632
-05130
180
-005556
07454
390
•002564
-05064
185
-005405
07352
400
•0025
-05
190
■005263
07255
420
■002381
-04880
195
■005128
07161
440
-002273
-04767
200
-005
07071
460
■002174
-04663
210
■004762
06901
480
■002083
•04564
220
-004545
06742
500
-002
•04472
Note to table xxviii.— This table shows values of V for given values of
CV-Kaud sJS.
The first line of the heading shews — , the third line JS. The figures in
brackets show the amount by which _- must be altered to alter JS and V
o
by 1 per cent. Thus for S=iv5-Vir t^^ slopes ^^^ and -jifrT give V 1 per cent.
more or less than in the table. For CJR = \OS, Fis 2^32 and 2^28 feet per
second.
Slopes not given in table. — To find ;^<S or V see following examples : —
S=\ in 10 100 200 2,500 15,000 40,000 50,000
See 1 in 1,000 10,000
Multiply by 10 10
800
2
10,000 3,7.50 10,000 500
2 i i A
Also see above note to table xxviii. Thus for 1 in 10, ^JB is •3162.
168
HYDRAULICS
Table XXVIII. (See note on preceding page.)
Values of
Cv'M
600
660
600
660
700
760
800
900
(10)
(11)
(12)
(13)
(14)
(16)
(16)
(18)
■04472
•04264
•04083
•03922
■03780
•03652
•03636
8-54
■03333
100
4'47
4 '26
4-08
3-92
8-78
3-65
3-33
102
4-56
4-35
4-17
4 00
3-86
8-73
3-61
3 40
104
4-65
4-44
4-25
4 08
3-93
3-80
3-68
3-47
106
4-74
4-52
4-83
416
4 01
8-87
3^76
3-53
108
4-83
4-61
4-41
4-24
4-08
3-94
3 82
3 60
110
4-92
4-69
4-49
4-31
416
4-02
8-89
3-67
112
5-01
4-78
4-57
4-39
4-23
409
3 96
3 73
114
5-10
4-86
4-66
4-47
4-31
416
4 03
3 80
116
519
4-95
4-74
4-55
4-39
4-24
4-10
3-87
118
5-28
5 03
4-82
4-63
4-46
4-31
417
3-93
120
6-37
5-12
4-90
4-71
4-54
4-38
4-24
4-00
123
5-60
5-25
5-02
4-82
4-65
4-49
4 85
4^10
126
5-6:}
5-37
5-15
4-94
4-76
4-60
4-46
4-2(>
129
5-77
5-50
5-27
5-06
4-88
4-71
4-56
4-30
132
5-90
5-63
5-39
6-18
4-99
4-82
4-67
4-40
135
6-04
5-76
5-61
5-30
5-10
4-93
4-77
4-50
138
6-17
^■88
5-64
5-41
5-22
5 04
4-88
4-60
141
6-31
6 01
5-76
5-53
5-33
5 1. 3
4 99
4-70
144
6-44
6 14
5-88
5-65
6-44
5-26
sot
4-80
147
6-57
6-26
6-00
5-77
5-56
5-37
5-20
4-90
150
6-71
6-40
613
5-88
5-67
5-48
5-30
5^00
153
6.84
6-52 ,
6.25
6-00
5.-78
5-69
5-41
5-10
156
6-98
6-65
6-37
6-12
5-90
5-70
5-52
5-20
160
7-16
6-82
6.53
6-28
6 05
5-84
5 66
5-38
164
7-33
6-99
6-70
6-43
6-20
6-99
.•i-80
5-47
168
7-51
7-16
6-86
6-59
6-35
614
5-94
5-60
172
7-69
7-33
7-02
6-75
6-50
6-28
6 08
5-73
176
7-87
7-51
7-19
6-90
6-65
6-43
6-22
5 87
180
8-05
7-68
7-85
7-06
6-80
6-57
6-37
6 00
185
8-27
7-89
7-55
7-26
6-99
6-76
6-54
617
190
8-50
810
7-76
7-45
7-18
6-94
6-T2
6-33
195
8-72
8-32
7-96
7-65
7-37
712
6-90
6-50
200
8-94
8 -63
8-17
7-84
7-56
7-80
7-07
(1 -c:
205
8-17
8-74
8-37
8-04
7-75
7-49
7 -20
6-83
210
9-39
8-95
8-57
8-24
7-94
m
7-4:i
7-00
216
9-62
9-17
8-78
8-43
813
7-85
7-60
7-17
220
9-84
9-88
8-98
8-63
8-32
8-08
7-7.S
7-33
225
10-1
9 '59
9'19
8-82
8-51
8-22
7-96
7-50
230
10-3
9-81
9-39
9-02
8-69
8-40
813
7-67
235
10-5
10-0
9-60
9-22
8-88
8-58
8-31
7-83
240
10-7
10-2
9-80
9-41
9 07
8-77
8-49
8-00
240
11-0
10-5
10-0
9-65
9-30
8-98
8-70
8-20
252
11-8
10-8
10-8
9-88
9-58
9-20
8-91
8-40
258
11-5
11-0
10'5
10-1
9-75
9-42
9-12
8 60
264
11 '8
11-3
10-8
10-4
9-98
9-61
9-34
8-80
270
12-1
11-5
11-0
10-6
10-2
9-86
9-55
9-00
276
12-8
11-8
11-8
10-8
10-4
10-1
9-76
9-20
282
I'J-O
12-(l
11-5
111
10-7
10-3
9^97
9-40
288
12-9
12-H
11'8
11-8
10-9
10-5
10-2
9-60
294
13-2
12-r,
I'J-O
11-5
111
10-7
10-4
9-80
800
13-4
12-8
12-3
11-8
11-3
11
10'6
10-0
PIPES
Table XXVIII. — Continued.
169
Values
1,000
1,100
1,200
1,300
1,400
1,500
1,600
1,800
2,000
of
(20)
(22)
(24)
(26)
(28)
(30)
(32)
(86)
(39-41)
CVR
•03162
•03016
■028S7
■02774
■02673
•02682
•02500
•02367
•02236
100
3-16
3^02
2^89
2-77
2-67
2^58
2-50
2^36
2^24
102
3-23
3-08
2
95
2
83
2
73
2
68
2
55
2-40
2.28
104
3-29
3^14
8
00
2
89
2
78
*)
68
2
i:0
2^45
2^33
106
3-35
3-20
3
06
2
94
'J
83
2
74
2
65
2^50
2-37
108
3-42
3-26
8
12
3
00
2
89
2
79
2
70
2-55
2 42
110
8-48
3-82
8
18
3
05
2
94
2
84
2
75
2-59
2-46
112
3-54
3-39
3
23
3
11
2
99
2
89
2
80
2 •64
2 51
114
3-60
3-44
3
29
3
16
8
05
2
94
2
85
2^69
2'55
111)
8-67
8-50
8
35
3
22
3
10
3
00
2
90
2^73
2-69
118
3-73
3-56
3
41
3
27
3
15
8
05
2
95
2-78
2^64
i:;o
3-79
3-62
3
46
8
33
3
21
3
10
3
00
2-83
2^68
123
3-89
8-71
3
55
3
41
3
29
3
18
3
08
2-90
2^75
126
3-f'8
8-80
3
64
3
50
3
37
3
25
3
15
2^97
2^82
129
4-08
3-89
3
71
3
58
3
45
3
33
3
23
3^04
2^88
132
4-17
3-98
3
81
8
66
8
53
3
41
3
30
3^H
2-95
135
4-27
4-07
3
90
3
74
3
61
3
49
3
38
3-18
8-02
138
4-86
4-16
3
98
3
88
3
69
3
56
3
45
3^25
3^09
141
4-46
4-25
4
07
3
91
4
77
8
64
3
53
3-32
3^15
144
4-55
4-84
4
16
4
00
3
85
3
72
3
60
3-39
3^22
147
4-65
4-43
4
24
4
08
3
93
3
80
3
68
3-47
3-29
150
4-74
4-52
4
38
4
16
4
01
3
87
3
75
3 '54
8-35
163
4-84
4-61
4
42
4
24
4
09
3
95
3
83
3-61
3-42
156
4-93
4-70
4
50
4
38
4
17
4
03
3
I'O
3-68
3^49
160
5-06
4-83
4
62
4
44
4
23
4
13
4
00
3-77
3^68
164
5-19
4-95
4
73
4
55
4
38
4
23
4
10
S-h7
3^67
168
5-31
5-07
4
85
4
66
4
49
4
34
4
■20
8-96
3^76
172
5-44
5-19
4
97
4
77
4
60
4
44
4
30
4-05
3^85
176
5-56
5-31
6
08
4
88
4
70
4
54
4
40
4-15
3^94
1?0
5-69
6 •4-'?
5
20
4
99
4
81
4
61
4
50
4-24
4-03
185
5-85
5-58
5
34
5
18
4
95
4
74
4
63
4-36
4^14
190
6 01
5-73
5
49
5
27
5
08
4
91
4
75
4-48
4^25
195
6-17
5-88
5
63
5
41
5
21
5
04
4
88
4-60
4-36
200
6-32
6-03
5
77
5
55
6
35
5
16
5
00
4-71
4-47
205
6-48
6.18
5
93
5
69
5
48
5
29
5
13
4-83
4^58
210
6-64
6-33
6
06
5
83
5
61
5
42
5
25
4 95
4^70
215
6-80
6-48
6
21
5
96
5
75
6
55
5
38
607
4^81
220
6-96
6^63
6
35
6
10
6
88
5
68
5
50
5-19
4-92
225
711
6-78
6
50
6
24
6
02
5
81
6
63
5 30
5^0:i
230
7-27
6-94
6
64
6
38
6
15
5
94
5
75
5-42
5^14
235
7-43
7 09
6
78
6
52
6
28
6
07
5
88
5-54
5^26
240
7-59
7-24
6
93
6
66
6
'2
6
20
6
00
5-B6
5-37
246
7-78
7 •42
7
10
6
82
6
58
6
35
6
15
5-fO
5 •SO
252
7-97
7-60
7
29
6
99
6
74
6
51
6
30
5-94
5^64
258
8-16
7^78
7
45
7
16
6
90 ^
6
66
6
45
6-08
5^77
264
8-35
7-96
7
62
7
32
7
06
6
82
6
60
6-22
5^90
270
8-54
8-14
7
79
7
49
7
22
6
97
6
75
6-36
6-04
276
8-73
8-32
7
97
7
66
7
39
7
13
6
90
6-51
6-17
282
8-92
8-50
8
14
7
82
7
55
7
28
7
05
6^65
6^31
288
911
8^68
8
32
7
99
7
80
7
44
7
20
6-79
6^44
294
9-30
8-86
8
49
8
16
7
96
7
r,9
7
35
6^93
6^57
300
9-49
9-05
8^66
8-32
8-02
7^75
7-50
7-07
6^71
170
HYDRAULICS
Table XXYlll.—Contimed.
Values
2,200
2,400
2,700
3,000
8,800
3,600
4,000
4,500
6,000
of
(43.40)
(47-49)
(63-&0)
(69-01)
(65-67)
(71-73)
(79-81)
(89-91)
(9U-102)
Cy'J!
■02132
•02041
■01926
■01826
■01741
■01667
■01681
■01491
■01414
100
2-13
2-04
1-93
1-88
1-74
1-67
1-58
1-49
1-41
102
2-18
2-08
1^96
1-86
1-78
170
1-til
1-52
1-44
104
2-22
2 12
2-00
1^90
1-81
1-73
164
1-55
1--J7
106
2-26
2-16
2-04
1-94
185
1-77
1-68
158
1-50
108
2S0
2-20
2-08
4-97
1^88
1-80
1-71
1-61
1-53
110
2-35
2-25
212
2-01
1-92
1-83
1-74
1-U4
1-56
112
2-39
2.29
2-16
2 05
1-95
1-87
1-77
1-67
158
114
2-43
2-33
2^20
2 08
1-99
1-90
1-tO
1-70
1-61
116
2-47
2-87
2 23
2-12
2 02
1-93
1-83
1-78
1-64
118
2-62
2-41
2 27
2-16
2 05
1-97
1-87
1-76
1-67
120
2-56
2-45
2-31
2-19
2-09
2 00
1-90
1-79
170
123
2 62
2-51
2^37
2^24
2-14
2-05
1-44
1^83
174
126
2-69
2-57
2-43
2 30
2^19
210
1-99
1-88
1-78
129
2-75
2 '38
2-48
■23B
2-25
215
2 04
1-92
1-82
132
2-82
2-69
2-54
2-41
2-30
2-20
2 09
1-97
1-87
135
2-88
2-7t)
2^60
2^47
235
2 25
218
2 01
1-91
138
2-94
2-82
2 66
2-52
2-40
2 30
2-18
2 06
1-95
141
8 01
•:-88
2-72
2-58
2-45
2-36
2 23
210
1-99
144
3 07
2-94
2-77
2-63
2 51
2-40
2-28
215
2 04
147
3-13
3-00
2-83
2-68
2-56
2 45
2-32
2 19
2^08
150
3-20
8-06
2^89
2-74
2-61
2-50
2-37
2-24
212
153
3-26
312
2^95
2 79
2 66
2-55
2-42
2-28
216
156
3-33
318
3^00
2^85
2-72
2 60
2-47
2-33
2-21
160
8 -Jl
3-27
3 08
2^92
2^79
2^67
2-53
2-39
2-26
164
3-50
3-35
316
3 00
2^86
2-73
2-59
2-45
232
168
3 58
3-43
3-23
3 07
2-93
2-80
2-66
2-51
2-38
172
3-67
3-51
331
3-14
3-00
2-87
2-72
2-57
2-43
176
3-75
3-59
3 39
321
3^07
2 93
2-78
2^03
2-49
180
3-84
8-67
3-47
3-29
313
3 00
2-85
2-68
2-55
185
3-95
3-7'i
8 56
838
8 22
3-08
2-93
2-76
2-62
190
4 05
3-88
3-66
3-47
3-31
8^17
3 00
2-83
2 69
195
4-16
3-98
3-75
3-56
3-50
3 25
3 08
291
2-76
200
4-2fi
4-08
3-85
3 65
8-48
3-33
3 16
2-98
2-83
205
4-37
4-18
3-95
3-74
8 57
3 -J 2
3 24
3-06
2-90
210
4-48
4-29
4-04
3^84
3-66
8 ■.Ml
3-82
313
2-97
215
4-58
4-89
414
3-93
3-74
3-58
3-40
3-21
3 04
220
4-69
4-49
4-24
4 02
3 88
8-67
8-48
3-28
311
225
4-80
4-59
4 33
411
3-92
3 75
8-56
3 -36
3 18
230
4-90
4-69
4-43
4 20
4-00
3-83
3 64
3-43
3-25
235
5 01
4-80
4-52
4-29
4 09
3 92
3-72
8-50
3 82
240
5-12
4 '90
4-62
4-88
4-19
4 00
3-79
3-58
3 39
246
5-25
5-02
474
4-49
4 28
410
8-89
3-67
8-48
252
5-87
5-14
4 85
4-60
4-89
4 20
8-99
3-76
3-56
258
5-50
6-27
4-97
4-71
4-49
4-30
4 08
8 -85
3 65
264
5-68
5 '39
5-08
4 ■,'•2
4-00
4-40
4-17
8-94
3-73
270
5-76
5 ■61
5-20
4-98
4 TO
4-50
4-27
4 08
3-82
276
6 '88
fi-C3
5-31
6 04
4-81
4 60
4-36
4 12
3^90
2-2
6-01
5'76
5 43
6-16
4-91
4-70
4-46
4-20
3 99
288
614
5 '88
5-54
6-26
6-01
4-80
4-55
4 29
4 07
2!ll
6-27
6 00
6-66
5^37
512
4^90
4-66
4^38
4-16
300
0'40
0-12
5-78
5-48
5^22
6^00
4-74
4-47
4 24
PIPES
171
Table XXVIII. — Continued.
Values
6,600
6,000
6,500
7,000
7,600
8,000
8,500
9,000
10,000
of
(lOS-112)
(118-122) (128-132)'(I38-142);
^148-162) (168-162)
(167-178)
(177-183)
(197-203)
C</R-
■01349
•01291 1
1
■01240 1
•01196
•01155
■01118
■01085
•01054
■0100
100
1-35
1-29
1-24
1-20
1-16
1-12
1-09
1-05
1-00
102
1-38
1-32
1-47
1-22
1-18
1-14
111
1-08
102
104
1-40
1-34
1-29
1-24
1-20
116
113
MO
1-U4
106
1-43
1-37
1-32
1-27
1-22
1-19
1-15
l-l:^
1-06
108
1 46
1-39
1-34
1-29
1-25
1-21
1-17
114
1-08
110
1-48
1-42
1-36
1-32
1-27
1-23
1-19
1-16
1-10
112
1-51
1-45
1-39
134
l--'9
1-25
V22
118
112
114
1-54
1-47
1-41
1-36
1-32
1-27
1-23
1-20
114
116
1-57
1-50
1-44
1-39
1-34
1-30
1-26
1-22
1-16
118
1-59
1-52
1-46
1-41
1-36
1-32
1-28
1-24
1-18
120
1-62
1-55
149
1-43
1-39
1-34
1-30
1-27
1-20
123
1-66
1-59
1-53
1-47
1-42
1-38
1'34
1-30
1-23
126
1-70
1-63
1-56
1-51
1-46
1-41
1-37
1-33
1-26
129
1-74
1-67
1-60
1-54
149
1-44
1-40
1'36
1-29
132
1-78
1-70
1-64
1-58
1-53
1-48
1-43
1-39
1-32
135
1-83
1-74
1-66
1-61
1-56
1-51
1-47
1-42
1-35
138
1-86
1-78
1-71
1-65
1-59
1-54
1-50
1-45
1-38
141
1-90
1-82
1-75
1-69
1-63
1'58
1-53
1-49
1-41
144
1-94
1-86
1-79
1-72
1-66
1-61
1-56
1-52
1-44
147
1-98
1-90
1-82
1-76
1-70
1-64
1-60
1-56
1-47
150
2-02
1-94
1-86
1-79
]-73
1-68
1-63
1-58
1-50
153
2 06
1-98
1-90
1-83
1-77
1-71
1-66
1-61
1'53
156
2-11
201
1-93
1-87
1-80
1-74
1-69
1-64
1-56
160
216
2 07
1-98
1-91
1-85
1-7^
1-74
1-69
1-60
164
2-l!l
212
2 03
1-96
1-89
1-83
1-78
1-73
1-64
168
2-27
2-17
2-08
2-01
1-94
1-88
1-82
1-77
1-68
172
2-32
2-22
2-13
2-06
1-99
1-92
1-87
1-81
1-72
176
2-37
2-27
2-18
2-10
2 03
1-97
1-91
1-86
1-76
180
2-43
2-32
2-23
2-15
2-08
201
1-95
1-90
1-80
185
2-50
2-39
2-30
2-21
2-14
2-07
2-01
1-95
1-85
190
2-56
2-45
2 -.36
2-27
2-20
2-12
2-06
2-00
1-90
195
2-63
2-52
2-42
2-33
2-25
2-18
2-12
2-06
1-95
200
2-70
2-58
2 48
2-39
2-31
2-24
2-17
211
2-00
205
2-77
2-65
2-54
2-45
2-37
2-29
2-22
2-16
2-05
210
2-83
2-71
2-60
2-51
2-43
2-85
2-28
2-21
210
215
2-90
2-78
2-67
2-57
2-48
2-40
2-33
2-27
2-15
220
2-97
2-84
2-73
2-63
2-54
2-46
2-39
2-32
2-20
225
3-04
2-91
2-79
2-69
2-60
2-52
2-44
2-37
2-25
230
3 10
2-97
2-85
2-75
2-66
2-57
2-50
2-42
2-30
235
3-27
3 03
2-91
2-81
2-72
2-63
2-55
2-48
2-35
240
3-24
310
2-98
2-87
2-77
2-68
2-60
2-53
2-40
246
3-32
318
3-05
2-94
2-84
2-76
2-67
2-59
2-46
252
3-40
3-25
313
3 01
2-91
2-82
2-74
2-66
2-52
258
3-48
3-33
3-20
3-08
2-98
2-88
2-80
2-72
2-58
264
3-56
3-41
3-27
316
3 05
2-95
2-86
2-78
2-64
270
3-64
3-48
3-35
3-23
3-12
3-02
2-93
2-85
2-70
276
3-72
3-56
8-42
3-30
3-19
3-09
3 00
2-91
2-76
282
3-80
3-64
3-50
3-37
3-26
315
3 06
2-97
2-82
288
3-89
3-72
3-57
3-44
3-33
3 22
313
3 '04
2-88
294
3-97
3-80
3-65
3-51
3-40
3-29
3-19
3
2-94
300
4 05
3-87
3-72
3-59
3-47
3-35
3-26
3-16
3-00
CHAPTEK YI
OPEN CHANNELS— UNIFOEM FLOW
[For preliminary information see chapter ii. articles 8-16 and 22-24]
Section I. — Open Channels in General
1. General Remarks. — Uniform flow can take place only in a
uniform channel. Strictly speaking, a uniform channel is one which
has a uniform bed-slope, and all its cross-sections equal and similar;
but if the cross-sections, though differing somewhat in form, as in
Fig. 98, are of equal areas and equal
wet borders, the channel is to all intents
and purposes uniform, provided the form
_---'^ of the section changes gradually. The
term ' uniform channel ' will be used in
Pig. 93. . IT..
this extended sense. ^ Breaches of uni-
formity in a channel may be frequent, and the reaches in which
the flow is variable may be of great length. The flow in a uni-
form channel is thus by no means everywhere uniform. Bends
are for convenience treated of in chap, vii., but flow round a bend
may be uniform. Thus a uniform stream need not be assumed to
be straight. It will be seen hereafter (chap. vii. art. IG) that
nearly everything which is true for uniform flow is true, with
some modifications, for variable flow.
The mean depth D (Fig. 99) of a stream is the sectional area
Q jj. A divided by the surface-width W
Since ^l=Dir=IlB, therefore the
hydraulic radius is less than the mean
^^ — depth in the same ratio as the surface-
''""'■ width is less than the border. This
will often assist in forming an idea of the hydraulic radius. The
greater the width of a stream in proportion to its depth, and the
' If 7i' varies in the opposito manner to S the flow may be uniform in a
variable channel, but this is very rare.
)7-J
OPEN CHANNELS — UNIFORM FLOW 173
fewer the undulations in the border, the more nearly will the
surface-width approach to the border and the hydraulic radius to
the mean depth. If the depth of water in a channel alters, the
hydraulic radius alters in the same manner. When the water-
level rises A increases faster than W, and B therefore increases ;
W
hut -=. decreases (unless the side-slopes are flat), so that B
increases less rapidly than D. For small changes of water-level
R and D both change at about the same rate.
2. Laws of Variation ofVelocity and Discharge. — For orifices,
weirs, and pipes it was possible to describe in a few words the
general laws according to which the velocities and discharges
vary, but for open streams it is not so. One law is simple, and
that is, that for any channel whatever V and Q are nearly as J S.
To double F" or Q it is necessary to quadruple S. For other
factors it is necessary to consider the shape of the cross-section.
For a stream of 'shallow section,' that is, one in which JV
greatly exceeds B, a change in W has hardly any effect on B or
on V, while Q is directly as JV. Also B is very nearly as D. For
depths not very small C is approximately as D'^, so that V is as
D'^. In this case, if D is doubled, Vis increased in the ratio 1-59
to 1. On comparing velocities, taken from tables, for channels
from 8 to 300 feet wide with sides vertical, or 1 to 1, and with
various velocities, the actual ratio is found to vary from 1-52 to
1'73. If the sides are steep A is nearly as D, and Q therefore as
jD^ or thereabouts. For a stream of 'medium section' — thatis,
one in which JV is 2 to 6 times D — with vertical sides A is as D,
and for moderate changes of water-level and depths not very small
V is nearly as B^, so that Q is as B^. Both these kinds of section
are extremely common. A flattening of the side-slopes may make
Q vary as B^^ If a stream has vertical sides and a depth far
exceeding its width — a rare case — the effects of JV and B are
reversed. For a triangular section — used for small drains — B is
as B, A as B"^, probably as B^, and Q as Z>».
For other kinds of section no definite laws can be framed, but
the effect of B is nearly always greater than that of JV, so that B
is the most important factor in the discharge, especially if the side-
slopes are flat, and S is always the least important factor.
If two streams have equal discharges, and have one factor in
the discharge equal, the approximate relation between the other
two factors can be found. Let two streams of shallow section
have equal slopes, and let one be twice as deep as the other. The
174 IIYDRAULIOS
latter must be (2)^ or 3-2 times as wide as the first. This law is
neaily the same as for weirs. When two reaches of a canal have
difl'erent bed-slopes, but equal and similar cross-sections, the
depth of water is, of course, less in the reach of steeper slope.
If the discharge is approximately as S^I)^, the depths in the two
reaches will be inversely as the fourth roots of the slopes. The
velocities are inversely as the depths, and are, therefore, as the
fourth roots of the slopes. A change of 40 per cent, in the slope
will cause a change of only about 10 per cent, in the velocity,
and a change of the same proportion, but of opposite kind, in the
depth of water. When the changes in the two factors are
relatively small they are inversely as the indices in the formula.
Suppose a stream of medium section with depth D and slope S
gives a certain discharge Q. Let X* be increased by a smaU
D 3S
amount — . Then the compensatory change in S will be — .
This principle may be applied in designing a channel to carry a
given discharge, whenever for any reason it becomes necessary to
make a slight change in the value first assumed for any factor.
The discharging power of a stream can be increased by in-
creasing the depth of water, the width or the slope, the last being
often effected by cutting off bends. The efficiencies of these
processes are in the order named. In any channel having sloping
sides both V and Q are more increased by raising the surface-level
than by deepening the bed by the same amount. It follows that
embanking a river is more effective than deepening it for in-
creasing its discharging power and enabling it to carry ofi' floods.
It is in fact the most efiective plan that can be devised.
In clearing out the head reaches of Indian inundation canals- —
so called because they flow only for a few months, when the rivers
are swollen — it used to be the custom to place the bed rather high,
at the ofT-take, in order to obtain a good slope. Of late years it
has been the custom to lower the bed, giving a flatter slope but a
greater depth of water. The velocity is about the same in both
cases, the increase in depth making up for the decrease in slope,
but the lowered bed of course gives a greatly augmented discharge.
On the other hand, the lowered bed must cause the introduction
of water moi'e heavily charged with silt. Moreover, the ratio of
depth to velocity in the canal is greater than before, and this
(chap. ii. art. 23) tends to cause increased deposit. Under the old
system of high beds the heads of the canals silted more or less.
It has Leon impossible to find out whether more silt has actually
opb;n channels — uniform flow 175
deposited since the introduction of the low-level system, because,
owing to changes in the course of the river, the same head channel
is seldom cleared for several years in succession, and also because
the quantity of silt deposited depends on other factors, such as the
position of the head, a canal taken off from the highly silt-laden
main stream silting more than one taken from a side channel.
Obviously the tendency of the low bed is to silt more than the
high one, but the worst that can happen is its silting up till it
assumes the level of the high one. This takes time, and while it
is going on an increased discharge is obtained.
Section II. — Special Forms of Channel
3. Section of 'Best Form.'^A stream is of the 'best form'
when for a given sectional area the border is a minimum, and the
hydraulic radius, therefore, a maximum. The velocity and dis-
charge are greater than in any other stream of the same sectional
area, slope, and roughness. The form which complies with this
condition is a semicircle whose diameter coincides with the line
of water surface. This form is used in concrete channels, but not
often in others, because of the diificulty of constructing curved
surfaces. Of rectilineal figures the best form is half a regular
polygon. The greater the number of sides the better, but in
practice the form of section is usually restricted to that having a
bed level across and two sides vertical or sloping. The best form
for vertical sides is the half-square (Fig. 100), and for sloping
sides the semi-hexagon (Fig. 101). If the angle of the side-
slopes is fixed (as it generally is) at some angle other than 60°,
Flc. 100. Via. 101. FlQ. 102.
the best form is that in which the bed and sides are all tangents
to a semicircle (Fig. 102). The bed-width is D{ Jn' + l—n), where
n is the ratio of the side-slopes. In every channel of the best
form the hydraulic radius is half the depth of water, and if the
section is rectilineal, the surface-width is equal to the sum of the
two slopes, so that the border is the sum of the surface and bed
widths.
The following statement shows the sectional areas of various channels of
the best form. All the channels have the same central depth D, the same
hydraulic radius — , and therefore the same velocity.
176
HYDRAULICS
Ratio of
Sectional
Description of Cross-section.
Sectional
Area.
Area to that
of the In-
scribed
Semicircle.
Semicircle
1-57 -D^
1-00
Half-square, . ...
2 D'-
1-27
Semi-hexagon
1-732D"'
1-10
Trapezoid, side-slopes 4 t" Ij •
1-7362)=
1-8282)=
1-11
1-16
H„ 1, ■ • •
2-106XI2
1-34
2 1...
2-4722)=
1-57
;, „ 3 „ 1, . . .
3-32623=
2-12
A channel of the best form is not usually the cheapest. If made of iron,
wood, or masonry the cost will probably be reduced by somewhat increasing
the width and reducing the depth, thereby enabling the sides to be made
lighter, though the length of border is slightly increased. In an excavated
channel, where the water-surface is to be at the ground-level, the best form
will give the minimum quantity of work and will be the cheapest if the
material excavated is rock, but if it is earth an increase of width and
decrease of depth will reduce the lift of the earth, and therefore the cost.
If the water-surface is not to be at the ground-level the cheapest form may
diflfer greatly from the best form.
If it is desired simply to deliver a stream of water of given
discharge with as high a velocity as possible, the best form is
suitable. If the object is to obtain high silt>supporting power,
so that the channel may not silt or may scour and enlarge itself,
the question of ratio of depth to velocity must be taken into
account ; and even when the object is to discourage the growth of
weeds the question of depth comes in.
If the depth of water in a channel fluctuates, the section can,
of course, be of the best form for only one water-level. Sewers
are often made of oval sections in order that the stream may be
of the best form, or nearly so, when the water-level is low, the
obj ect being to pre ven t deposi ts. In
Fig. 103 (Metropolitan Ovoid) the
radius of the invert is half that of
the crown, and in Fig. 104 (Hawkes-
ley's Ovoid) nearly three - fifths.
There is also a form known as
Jackson's Peg-top Section. In each
case the velocity with the sewer one-
third full is about three-fourths of the velocity when it is two-
thirds full.
Fio. 108.
Pio. 104.
OPEN CHANNELS — UNIFORM FLOW 177
4. Irregular Sections. — The cross-section of a stream may be
called 'irregular' -when the border contains undulations or
saliences of such a character as to divide the section into well-
marked divisions (Fig. 105).
In this case the vi^ater in each it ] i •'' ^~^
division has a velocity of its \\ | | V. ^1
own, and in order to calculate -^ ' \|~ --^--y^a d
the discharge of the whole ^^ '^
stream by the use of the
formula V=C J US, it is necessary to consider each division
separately, finding its hydraulic radius from its area and border.
The length AB is not included in the border of either division,
since if there is any friction along it, it accelerates the motion in
one division and retards it in the other. If A^ A^ are the
sectional areas, and E^ R^ the hydraulic radii,
Q, = C,A,JU^
Q, = C,A,jB,S.
The discharge of the whole channel, calculated from the equation
Q=CA sj US, equals Qi + Q^ only when i^, = itj, otherwise it is less.
The more U^ and R^ differ, the more Q diners from Q1 + Q2, and
for given values of Ri and R^ the difference is greatest when
Ai = Ai. If either Ai or A« is relatively very small, the diflferenee
between Q and Q^ + Q„ will be small. It may happen that iJ, and
i?2 differ greatly with low supplies, and not much with high
supplies. If without altering either the length of the border or
the sectional area of the stream the border be changed to CDEF,
the section is no longer irregular, and the equation V~ J US is
the proper one to use. There are thus two cross-sections with
equal values of R and different mean velocities, that is, different
values of 0. Even in a regular section the same principle holds
good. The discharge is the sum of the discharges of a number
of parts, and may be affected by a change in the form of the
border alone. (See also art. 13.)
An instance of an irregular section occurs when a stream over-
flows its banks (Fig.
106). As the overflow
occurs the border of
the whole stream may
increase far more
Fig. 100.
rapidly than the sec-
tional area, and Q, if calculated as a whole, would diminish with
rise of the water-level. The velocity and discharge of the main
M
178 HYDRAULICS
body and of the overflow must be considered separately, and both
will increase as the water-level rises. Similarly, if there are
longitudinal grooves or ruts in the bed of a stream, such, for
instance, as those caused by longitudinal battens, the water in
the grooves has a separate velocity of its own, and the velocity of
the main body cannot be reduced indefinitely by increasing the
number and depth of the grooves, although the border can be
increased in this manner to any extent. If the river is winding,
the spill-water, which flows straight, may have a slope greater than
that in the river channel, but its velocity may still be very low,
especially if the country is covered with crops or vegetation.
Some of the spill-water, however, disappears by absorption, and
it is clear that in every case it takes off some of the discharge
of the river. Thus the embanking of a river, so as to shut off
spills, must necessarily, to start with, raise the flood-level.
Whether scour of the channel subsequently reduces the level is
another matter.
5. Channels of Constant Velocity or Discliarge. — Let A be the
area, B the border, and W the surface-width of any stream
whose water-level is JiS (Fig.
^^"-'^ 107), and let the water-level
rise to TU, the increase in depth
being a small quantity d and
the increase in the surface-width
j,jg jQ^ being 2w. Then if the slopes
BT, SU be made such that
{TF+w)d _J
1^ s— ij! the border will have increased in the same ratio
as the area, and R will be unaltered. By using the new values
of A and B, corresponding to the raised surface, the process can
be continued, but the slope becomes rapidly flatter. If the
surface falls below RS; R is no longer constant, but decreases.
It is impossible to design a section such that R will remain
constant as the depth decreases to zero. And OAcn within the
limits in which ]l is constant, the mean velocity is not constant.
The channel is irregular, and the velocity, both in the main body
of water and in the minor ones, increases as the water-level rises.
The investigations which have at times been made to find the
equation to the curve of the border when R is constant are
useful only 'as mathematical exercises.
The velocity as the water-level rises is nearly constant in a
very deep, narrow channel with vertical sides, and it may be kept
OPEN CHANNELS — UNIFORM FLOW
179
quite constant by making the sides overhang — as in a sewer
running nearly full — but the process is speedily terminated by
the meeting of the two sides.
To keep the discharge constant for different water-levels is still
more difficult, but would be of great practical use, especially in
irrigation distributaries. It could be effected, by making the
sides overhang, but they would have to project almost horizontally
and would very soon meet, thus giving only a small range of
depth. Any form of section adopted for giving either constant
velocity or constant discharge must be continuous along the
channel from its head for a great distance. If of short length
the slope or hydraulic gradient in it would be liable to vary
greatly, and with it the velocity. (Cf. chap. ii. art. 14.)
6. Circular Sections. — A channel of circular section is an open
channel when it is not running full. In such a channel the
hydraulic radius, and therefore the velocity, is a maximum when
the angle subtended by the dry portion of the border is 103°, or
the depth is "81 of the full depth. If the depth is further
increased R decreases, but at first the increase of area more
than compensates for this, and the discharge goes on increasing.
When the angle above-mentioned is about 52°, or the depth is -95
of the full depth, the expression AG JR is a maximum, and Q is
then about 5 per cent, more than when the channel is flowing full.
Section III. — Relative Velocities in Cross-section
7. G-eneral Laws. — Except near abrupt changes the water at
every point of a cross-section of a stream has its chief velocity
parallel to the axis of the stream and in the direction of flow,
and the velocity varies gradually from point to point. Although
the velocity at any point in a cross-section is affected to some
extent by its distance from every part of the border, it depends
chiefly on its distance ^
from that part of the m~
border which is near to
it. Those portions of
the border which are
remote from the point
have a small, often an ^^^ ^^^
inappreciable effect. In
Fig. 108 the velocity at A is less than at B because of the effect
of the neighbouring side. At all points between G and D the
180 HYDRAULICS
velocities are nearly equal because both sides are remote.
Given the cross-section of a stream, the forms of the velocity
curves are known in a general way but not with accuracy. In
other words, their equations are not known.
The law that the velocity is greatest at points furthest from
the border is subject to one important exception. The maximum
velocity in any vertical plane parallel to the axis of the stream
is generally at a point somewhat below the surface and not at
the surface. If D is the depth of water and Dm the depth of the
point of maximum velocity, the ratio -jr- in a stream of shallow
section at points not near the sides may Have any value from
zero to "30, and if the side-slopes are not steep the same ratio
may be maintained right across the channel. When the sides
are very steep or vertical the ratio —^- close to the side is about
■50 or '60, and it decreases towards the centre of the stream,
attaining its normal value in a shallow section at a distance from
the side equal to about W or l-oD, and thereafter remains constant
or nearly so.
The depression of the maximum velocity has been sometimes
attributed to the resistance of the air, but this theory is now
quite discredited. Air resistance could cause only a very minute
depression, and it cannot account for the variation of the
depression at different parts of a cross-section. It is true that
wind acting on waves and ripples may produce some effect.
The water-level in the Red Sea at Suez is raised during certain
seasons of the year when the wind blows steadily up the Red Sea.
On the Mississippi, with depths ranging from 45 to 110 feet, an
upstream wind was found to reduce the surface velocity and
Dm
increase the ratio -jr. A downstream wind produced opposite
effects, but even with a downstream wind the maximum velocity
was below the surface, and the same thing has been observed
elsewhere. Wind acting on ripples '^ is a different thing from simple
air resistance. The depression is attributed by Thomson to the
eddies which rise from the bed to the surface. The water of
which the eddies are composed is slow-moving, and though the
eddies retard the velocity at all points which they traverse,
thoy have most effect at the surface, because they spread out and
accumulate there. This explanation seems to be the true one,
at least as regards the central portions of a stream. When no
* Wind which produces waves can cause currents in large bodies of water.
OPEN CHANNELS — UNIFORM FLOW
181
depression exists there, it is because the eddies are weak relatively
to the other factors. The increased depression of the maximum
velocity near the sides vv^hen these are steep or vertical is clearly
connected with certain currents which circulate transversely in a
stream. Near the side there is an upward current (Fig. 108), at
least in the upper portion of the section, and there is a surface
current from the side outwards. It is this current which causes
floating matter to accumulate in mid-stream. At a lower level
there must be an inward current which brings quick-moving water
towards the sides, while the slow-moving water near the surface
travels outwards and reduces the surface velocity at all points
which it reaches.
As to the cause of the currents, Stearns, who has investigated
the subject, '^ considers that they are due to eddies produced at
the sides. The eddies from the side tend on the average to move
at right angles to it, but they also tend
to move chiefly in the direction of the
least resistance, that is, towards the
surface.
8. Horizontal Velocity Curves. — A
horizontal ' mean velocity curve ' is one
whose ordinates are the mean velocities
on difierent verticals extending from
surface to bed. The general forms of
these curves for a rectangular section
are shown in Fig. 109 for two water-
FlG. 109.
levels. When the section is shallow
the velocities on different verticals, at a distance from the side
exceeding iD or
3i), become near-
ly equal. Fig.
110 show;s a
channel with
sloping sides.
The length in
which the velo-
city is practically
constant is some-
Fio. no. yvh&i greater
than before, and
the curves in this portion are nearly as before, but the part in
' Transactions o/ihe American Society of Civil Engineers, vol. xii.
182 HYDRAULICS
which the velocity varies is longer, both actually and relatively to
the whole width. If the bed is not level across (Fig. 105, p. 177)
the velocity is greater where the depth is greater. If there are,
at a distance from the sides, divisions of considerable width and
constant depth, as HG and BK, the velocity in each such division
is nearly constant. The rough rule for a channel of shallow
section considered as a whole, that T^is approximately as i>* where
D is the mean depth, probably applies to any two divisions such
as those under consideration and to the same division for different
water-levels. But if a division is of small width its velocity is
affected by those adjoining it. The velocity at B is affected by the
greater velocity between B and G. This, combined with the fact
that V is approximately as 2)1, causes the velocity curve to be one
which tones down the irregularities of the bed. On the South
American rivers with depths of 9 to 73 feet, gradually increasing
from the bank to the centre of the stream, Revy found the velocity
to vary as 2)" where n is greater than unity, but this conclusion
appears to be unsound.^ The form of the velocity curves in a
channel of irregular sections changes, as it does in regular channels,
with the water-level. Irregularities which have a marked effect at
low water may have no perceptible effect at high water.
The nature of the horizontal mean velocity curve depends on
the shape of the cross-section, and not on its size. From observa-
tions made by Bazin on small artificial channels lined with plaster,
plank, or gravel, with widths of about 6-5 feet, and depths up to
1-5 feet, and observations made by Cunningham on the Ganges
Canal in an earthen channel about 170 feet wide and 5 feet deep,
and in a masonry channel 85 feet wide, with depths of 2 feet to
3-5 feet, it is also proved that if the velocity is altered by altering
the surface-slope (and in the case of Bazin's channels by altering
the roughness), the velocities on different verticals all alter in
about the same proportion. It is probable, considering the com-
plications arising from eddies and transverse currents, that the
actual size of the channel has some effect, but it is negligible, at
least in streams of shallow section, and under the conditions
which occur in practice.
Let U be the mean velocity on the central vertical, and V that
in the whole cross-section. Let jy—'^- The values of the co-
efficient a arc as follows : —
' Si'e Notes at end of chapter.
OPEN CHANNELS — UNIFORM FLOW 183
^'fdeptr'" ^''^*^} ^ 1-5 2 3 4 5 6 7 10 20 30 50 90
Value of a, . . -SC -87 -88 -89 "90 -91 '92 -93 -94 -95 -96 -97 '98.
These co-efScients are applicable to rectangular and trapezoidal
channels, but may not be very accurate for the latter when the
ratio of the mean width to the depth is small, especially if the
side-slopes are flat. In other cases they are probably correct
to within 1 or i) per cent, for the deeper sections, and to within
•5 per cent, for shallower sections. The co-efScients have been
found chiefly from the observations above mentioned. Bazin did
not work out this particular co-efEcient, but his figures enable it
to be found. In any particular channel the co-efficient increases
as the water-level falls.
The co-efficient a was determined in the observations on the
Solani aqueduct in the Roorkee experiments. In the aqueduct
there is a central wall which divides the canal into two channels,
each 85 feet wide. The aqueduct is 932 feet long, and the
observations were made in the middle, that is, only 466 feet from
the upper end. Upstream of the aqueduct the canal consists of
one undivided channel, and the greatest velocities are in the
centre. Owing to this fact the maximum velocities at the observa-
tion sites in the aqueduct at times of high supply are not in the
centres of the channels, but nearer the central wall.^ The velocities
observed to determine a were, however, made in the centres of the
channels, and the resulting values of a were therefore too high.
The depth varied from 4 to 10 feet, and the ratio of width to
depth therefore from 21 to 8'5. The values of a were nearly
constant at -95 or '96. For the lower depths the co-efficient
agrees with that in the above table. For the higher depths it was
overestimated for the reason just given. (See chap. ii. art. 21.)
The co-efficients are strictly applicable only when the bed, as
seen in cross-section, is a straight and horizontal line, but prac-
tically they are applicable whenever the central depth is the mean
depth (not counting the sections over the side-slopes), and does not
differ much from the others. If there is a shallow in the centre
the co-efficient may exceed 1 '0, and may increase greatly at low
water. For some particular sections somewhat hollow in the
centre the co-efficient may not vary as the water-level changes.
The above refers to horizontal mean velocity curves. The
properties of horizontal curves at particular levels, for instance at
the surface, mid-depth, or bed, are, generally speaking, similar to
the above. In the central portions of the stream the curves are
' Not at low supply, cf. notes on momentum at end of chapter vii,
184
HYDRAULICS
probably all parallel projections of one another. Near to vertical
or very steep sides, owing to the greater depression of the line
of maximum velocity, the mid-depth velocity curve, and to some
extent the bed-velocity curve, become more protuberant, and the
surface curve less so. Fig. Ill shows the distribution of velocities
Fia. 111.
found by Bazin in a channel 6 feet wide and I'o feet deep, linea
with coarse gravel. Each line passes through points where the
velocities are equal.
9. Vertical Velocity Curves. — The general forms of the curves
are shown in Figs. 112 and 113.i Many attempts have been made
Surface
Surface Telociiv
JBed Velocity
FlQ. 112.
Bed Velocity
Fio. 113.
to find the equations to the curves, and it is sometimes said that
the curve is a parabola with a horizontal axis corresponding to the
line of maximum velocity. This is improbable. The transverse
curve is certainly not a parabola. The bed of a channel retards
the flow in the same manner as the side retards it, and the velocity
probably decreases very rapidly close to the bed just as it does
close to a vertical or steep bank. Except near the bed, almost
any geometric curve can be made to fit the velocity curve. The
equation to the curve is not nearly of so much practical importance
as the ratios of the different velocities to one another. If these
are known, the observation of surface velocities enables the bed-
velocities and moan velocities to be ascertained. A slight differ-
ence in the ratios may make a great difference in the equation.
Even the information regarding the ratios is very imperfect, and
' The floats and dotted lines are referred to in chap. Wii.
OPEN CHANNELS — UNIFORM FLOW 185
until it is improved it is useless to discuss the equation. When
the depths on adjoining verticals are not equal, the curves are
probably of a highly complex nature, since each must influence
those near it.
Let Us, U,n, U, and U^ be the surface, maximum, mean and bed
velocities on any vertical not near a steep side of a channel,
then the ratios which are of most practical importance are those
of -Dm to D, and of U to each of the other velocities. The results
as to these ratios furnished by experiments show great discrep-
ancies. The fact seems to be that the ratios are easily disturbed.
A change in depth,^ roughness, or surface-slope may cause the
eddies to rise in greater or less proportion, and so alter the
ratios. The quantity of solids moved perhaps aflfects them, since
some of the work of the eddies is expended in lifting or moving
the materials. Wind may affect the surface velocities and un-
steadiness in the flow may affect the ratios.* The depth i)„ is
seldom accurately observed. This is because the velocities above
and below the line of 17^, differ very slightly from f „, and a,lso
because the velocities are not generally observed at close intervals.
A greater defect is in the observation of bed velocities. They are
seldom T)bserved really close to the bed. When so observed a
rapid decrease of velocity has been noticed.
Generally the different ratios roughly follow one another.
When the eddies reach the surface in greater proportion the ratio
-^ increases. At the same time U,n is diminished and C/j is
increased, because more quickly moving water takes the place of
that which rises. Thus the different velocities tend to become
equal and the ratios to approach unity. It will be sufficient to
D U*
consider for the present only the ratios -^ and -— . On examin-
ing the results of experiments no clear connection between these
ratios and the quantities U and D is apparent, but by considering
the two separate elements on which, for any given depth, U
depends, namely N and »S^, some more definite, though not very
satisfactory results are obtained. The following table contains
an abstract of the results of some of the chief observations. Each
group consists generally of several series, each series having a separ-
ate value of D and U, and sometimes of N or S. The table is a mere
abstract, and is intended to show only what experiments have
been considered and their general results. On the Mississippi and
Irrawaddy and Ganges Canal the observations were made with
' Changes in the bed often occur and may not be noticed.
' But see chap. ix. art. 5.
* Or rr-, which is nearly the same.
186
HYDRAULICS
Abstract of Results of Observations on Verticals
NOT NEAR the SiDES OF THE CHANNELS.
b
Depth, Roughness, and
Ratios. 1
if
Channel.
Observer.
Velocity on Vertical.
D™
n
K
D
N
u
D
Um
Division L— Great Rivers. -v-^. "^ p
1
Mississippi.
Humphreys
and Abbott.
76
•027
3-5
•38
•98
o
79
•027
21
•13
■94
3
65
•031
53
■27
■97
4
27
•025
4-7
■28
■97
5
Irrawaddy.
Gordon.
50
...
5^4
•03
■95
6
29
1-8
zero
■93
7
Parana de las
Palmas.
Revy.
50
24
zero
•83
8
La Plata
,,
24
\-3 1
zero
■69
Division 11. — Ordinary Streams.
9
Saone.
Leveill^.
14
•028
2"2
■15
■90
10
Garonne.
Baumgarten.
11
•0-275
5-0
■10
■90
11
Seine.
Emmery.
9
•026
2-5
■05
■89
12
Rhine.
International
Commission.
7
•030
7-1
zero
■85
13
Branch of Rhine
Defontaine.
5
■0275
3-5
zero
■87
14
Ganges Canal.
Cunningham.
9
•025
3-5
•12
■88
15
„
,,
6-5
•013
4^2
■19
93
Division III. — Small Streams.
iij
Artificial
Channels.
Bazin.
1-3
■020
^■9
■05
■84
17
>)
11
■015
66
zero
■89
18
1
•012
65
zero
•91
19
"
•9
•010
91
zero
•92
1
the double float, and the ratio was thus seriously vitiated (chap.
viii. art. 9), the values of U obtained being too high. On the
Ganges Canal U was, however, observed separately by means of
rod-floats, and by making certain corrections for the length of rod
used, corrected values of U have been found and used. In Revy's
observations the flow was unsteady.
By considering the figures of each separate series in divisions
OPEN CHANNELS — UNIFORM FLOW
187
ii. and iii. it is quite clear that the ratio jj- increases as N de-
creases. This result had previously been found by Bazin for his
small channels. The ratio also increases with the depth. In
division i. the figures are unreliable, as above explained, but to
some extent they confirm the above laws. From a consideration
of the various results the following table has been prepared. The
figures are an advance on the former rough rule that the ratio is
' -85 to '90.' The blanks in the table may be filled in according
to judgment. In some small and rough channels the ratio has been
found to be as low as '60. The ratio Yf inay be designated /3.
Probable Eatios of Mean to Surface Velocities /'yS or -^ j
ON Verticals not near the Sides of a Channel.
Values of N.
Depth on
Verticil.
•030
•0275
•02B0
•0225
•020
•01'
5 ■OlB
■013 ■
JIO
Feet.
•90
•83
•8
6 -88
•89 ■
91
1-0
•78
•82
*••
..
1-10
•84
•8
7 -89
•90 ■
91
1-25
•85
•8
7 ^89
•91 •
91
1-50
...
...
■87
•8
8 ^90
■91 ■
92
2-00
•80
•86
3-00
•83
•88
5-0
•85
■87
•89
•93
7-0
•90
100
•86
•89
■90
■92
13-0
•91
15-0
■87
•91
18-0
•88
•'91
■91
•b'i
. •bi
■gi
20-0
•88
•92
23-0
•'93
...
28-0
•95
...
After the preparation of the above table for depths up to 18 feet
the author's attention was drawn to an extensive and careful series
of observations made with current-meters by Marr on the Mississippi.^
The results worked up and abstracted are as follows : —
' Report on Cunent-meter Observations in the Mississippi, near Burlington.
The figures for depths of 1 5 and 20 feet have been obtained from Parker's Control
of Water. On the Irrawaddy (N not known) the average ratio was found to be
■89, •90, '93, and ■QT for average depths of about 53, 32, 64, and 3i feet respec-
tively. Individual observations showed great irregularities, e.g. the -QZ ratio
varied from 1 •01 to ^90 {Note on the Irrawaddy River, Samuelson, Govern-
ment Press, Rangoon).
188
Feet.
Depth=ll-2
Feet.
13-2
Fppt.
20'4
F<'et.
21^6
Feet.
27-6
r= 2-0
2-6
1-9
2-2
2-2
U^ V,= -89
•91
■93
•93
•945
i)„^/;= -09
•09
•2G
•21
■09.
The values of N and S are not stated, but N is judged to have
been about •0275, and the above table has been accordingly ex-
tended to depths of 28 feet. The velocities were not observed near
enough to the bed to enable U,, to be found.
When the maximum velocity is at the surface the ratio -=^
is the same as — . Otherwise it is 1 to 3 per cent, lower.
No law for the variation of -J? can be traced, except that in
small streams the ratio is greater the rougher the channel. The
ratio never exceeds ^20 except on the Mississippi. On the
Irrawaddy, with not dissimilar depths and velocities, it is very
small or zero. The difference may possibly be due to differences
in J\rand S. It appears that in very deep rivers all the ratios are
more sensitive.
The ratio
?/,•._. U, „.. ,_„ .^.. __.,. I
U^
or jY- generally follows the ratio jj-. In
the detailed series of division iii. of the table on page 168, both
ratios attain maximum and minimum values together. Values
ranging from ■58 to ■GS have been found for the ratio on the
Lower Ehine, Meuse, Oder, Worth, and Messel. It is probable
that in nearly all experiments the ratios found are too high
because the velocities are hardly ever observed close to the bed,
and also because of the rapid decrease of velocity near the bed.
On the S3,one the current-meter was placed as near to the bed as
possible, and the ratio comes out very low. The following table
shows such probable values of this ratio as it has been possible
to arrive at : —
i^
•080 ■027.i
•0J6
•020
•015
•010
Depths,
FOBt.
5 to 18
•50 to •55
Feet,
1 to 1^5
•50 to 55
Ftet.
1 to Ij
•60
Feet.
1
•65
When the various ratios are known the vertical velocity curve
OPEN CHANNELS — UNIFORM FLOW 189
can be drawn. The curves are, of course, sharper the less the
depth of water. The depth at which the velocity is equal to the
mean velocity on the vertical varies somewhat, being generally
deeper as i)„, is deeper. It has been found to vary from -SSI* to
■&7D. On the average it is at about -QOD or ■Q25D. The mid-
depth velocity is greater than the mean, but generally by only 1 or
2 per cent. On the Mississippi it was found to remain constant
while U was constant, even though U, was increased or decreased
by wind, a compensating change occurring near the bed. The
mean velocity can be found approximately by an observation at
about '60 of the full depth. It can be found very nearly, as has
been shown by Cunningham, by observing the velocities at '21
and '79 of the full depth and taking the mean of the two.
10. Central Surface Velocity Co-efficients. — Sometimes the
mean velocity F in a cross-section is inferred from an observation
in the centre of a stream. If U is the velocity on the central
vertical V=a.U. Sometimes U„ the central surface velocity, is
observed and multiplied by a co-efficient 8. It is clear that 6 must
be a X /?. It has been seen that a depends on the shape of the
section, and is practically independent of the size, roughness, and
slope, while yS, at least in streams of shallow section, seems to
depend on two of these factors. In a given stream of shallow
section and fairly level bed a decreases as D increases, but /3
increases. Hence S does not in ordinary cases show any very
great fluctuation. On the Ganges . Canal, with earthen channels
190 to 60 feet wide, and masonry channels 85 feet wide, and
with depths of water from 2 to 1 1 feet, 8 varied from -84 to '89.
Neither a nor fi varied much. With widths of 10 to 20 feet, and
depths of 1 to 3 feet, a was somewhat reduced, and 6 was also less,
its values being -81 to '85. At one site, where there was a shallow
in the middle, a rose at low water to 1 -07 and 8 to '95. Ordinarily
8 is seldom below -80.
Bazin found for small channels the values of a co-efficient A,
giving the ratio of C„ to V. Its values do not differ very much
from those of 8. Bazin, however, assumed that A depended only
on N and R, and on this assumption he worked out values of the
co-efficient for values of R, extending up to 20 feet, or far beyond
the limits of his experiments. It has been the custom to use these
co-efficients as values of 8, that is, to use them for obtaining V
from U,. This in itself would not cause any very large error, but
the values of the co-efficients, when applied to channels of slopes,
sizes, and roughnesses, differing greatly from those used by Bazin,
190
HYDRAULICS
are entirely wrong. Neither S nor A can depend only on B and N,
but must depend on the values of a and (i.
Other general expressions for 8 have been proposed by Prony
and others, but they, in common with those of Bazin, are almost
useless as general formulse.
Section IV. — Co-efficients
11. Bazin's and Kutter's Co-efficients. — Setting aside obsolete
and discarded figures, the first important set of co-efficients for
open channels is that obtained by Darcy and Bazin from experi-
ments on artificial channels, whose width did not exceed 6'56 feet
in masonry and wood and 21 feet in earth. Bazin, from these
experiments, framed tables of G (connecting them by an empirical
formula, and extending them far outside the range of the experi-
ments) for four classes of channel, namely, earth, rubble masonry,
ashlar or brickwork, and smooth cemented surfaces. It has been
found that these co-efficients, though correct enough for small
channels, often fail for others. More recently two Swiss engineers,
Ganguillet and Kutter, went thoroughly into the subject, and
after investigating the results of the principal observations, and
making some themselves, arrived at various sets of co-efficients for
channels of different degrees of roughness, the roughness being
defined by a 'rugosity-co-efficient' N. The following statement
shows some selected values of Bazin's and Kutter's co-efficients.
The last three columns will be referred to below : —
Hy.
draulic
Radius
Bazin's
Co-effieients.
Kutter's Co-efflcients for
Channels having a Slope of
1 in 5000.
Bazin's
New Co-eiBcients.
Cement,
etc.
Bubble
Masonry.
Earth.
Cement,
Plaster,
etc.
Earthen
CUaniiels
in Good
Order.
Earthen
Channels
in Bad
Order.
Cement,
etc
Regular
Channels.
Very
Rough
Channels.
iY= -010
^■=•020
.V= -OSO
•).= -109
y = l-54
V=S-17
■5
1-0
2-0
4-0
6(1
100
135
141
144
146
147
147
72
87
98
106
110
112
.16
48
62
76
S-t
01
132
ir.2
170
KS5
ins
201
57
09
82
94
101
108
35
43
53
63
69
76
136
142
146
149
151
152
50
60
75
89
97
106
29
36
49
61
69
79
It will be seen that (' always increases with R, and that the
increase is less rapid as li beconjes greater, and that as B increases
OPEN CHANNELS — UNIFORM FLOW
191
C becomes less affected by the degree of roughness. Also that,
with change of R, Kutter's co-efficient varies more than Bazin's for
smooth channels, and less than Bazin's for rough channels.
Bazin's co-efficients are independent of S, but Kutter's depend
to some extent on iS^, as will appear from the following statement : —
Value of &
Kutter's Co-elficients for differeut Slopes.
N= -010
JV= -030
Slope 1 in
10,000
Slope 1 in 1000
and Steeper
Slopes.
Slope 1 in
10,000.
Slope 1 in 1000
and Steeper
Slopes.
■5
1-0
2-0
4-0
6-0
10-0
126
148
168
186
195
206
138
156
172
185
191
197
33
42
52
64
70
78
30
'15
54
63
68
74
When R is about 3-2, C is independent of S. It increases or
decreases with S according as R is below or above 3-2, but it
varies only slightly for a great change of S, the variation being
greatest when S is between 1 in 2500 and 1 in 5000. For slopes
steeper than 1 in 100.0 the variation is negligible. For all values
of N the variation of C with S is very similar in relative amount.
Kutter's co-efficients for flat slopes are based on the Mississippi
observations of Humphreys and Abbott. The fall here was
small, sometimes only '02 foot per mile, and doubt has been
cast on the reliability of the slope observations. Bazin, who
subsequently reviewed the whole question and considered all the
best-known experiments, arrived at a new set of co-efficients,
some of whose general values are given in the last three
columns of the first of the above tables. As before, he
makes G independent of S, and his different sets of co-efficients
correspond to certain values of y which is analogous to 'Kutter's
N. The rate at which C varies with change of R conforms
more nearly than before to that of Kutter's co-efficients. Bazin
in his discussion includes some results which are known to
be wrong, such as those obtained on the Irrawaddy (art. 9)
and in the Solani aqueduct, Ganges Canal (chap. vii. art. 5),
but the rejection of these would not appreciably alter his figures.
192 HYDRAULICS
The question has recently been discussed by Houk ^ who con-
cludes that the Mississippi observation at Columbus and two of
those at Carrollton should be rejected — the fall in these cases
having been so small that S may easily have been from 55 to 161
per cent, in error, — but that in the case of the observations at
Vicksburg and two others at Carrollton the error would be, say,
7"5 to 27 per cent. He concludes that though it is not proved
that Kutter has ascertained the exact law, he is correct in making
C increase with decrease of S in deep streams, and he gives details
of subsequent observations on the Irrawaddy, Mississippi, Bogue
Phalia and Volga — with JR averaging 20 to 50 feet — all tending to
confirm this law. Considering all the information available, includ-
ing Bazin's figures and arguments, and the various formulae which
have been propounded, including some recent ones, Houk con-
cludes that the Bazin formula is inferior to Kutter's for all types of
open channels, and that although the Kutter formula is not ideal it
is the best available. This conclusion is accepted.
Manning adapts Kutter's co-efficients, by putting
N
C is independent oi S. It varies in the manner described for a
stream of shallow section (art. 2). Complete sets of Kutter's,
Bazin's, and Manning's co-efficients — Cjf, Cb, and Cj, — are given in
tables xxix. to xlii. A diagram (Fig. 113a) iS also given. In the
diagram there are shown, for various values of N, curves of Cj,
for slopes of 1 in 1000 and 1 in 20,000, by continuous lines,
and the curves of G^i by small dashes. The curves of Cb are
shown by longer dashes.
In Cfl the number of classes is far too small. C k is far more
used than either of the others. It is sometimes said to be com-
plicated, but this chiefly means that for one value of iV there are
six columns of figures. This causes little trouble when proper tables
or diagrams are used.
Small smooth open channels have been dealt with in chapter v.
art. 9. The rapid decrease of (.'k when It is small is there men-
tioned and dealt with, and it is stated that small channels are the
most sensitive to changes in roughness. This has doubtless been a
cause of the error in C,;. Tlxe best co-efficients for small open
channels — say R less than 2 foot, when iV is 'Oil — are probably
tliose in table xxvA.
' Calculation of Fluw in Open Channels. See chap. iv. art. 15.
OPEN CHAKNELS— UNIFOKM FLOW
193
For large smooth channels the difference between C^ and the
other co-efficients is often great. The number of such channels is
limited and the number of observations in them has been small.
The question is generally obscured by variations in the roughness.
In order to bring out the law of variation of C with R, observations
are required on the same channel with different depths of water.
These, are not often obtained. For the rougher channels — these
are generally channels in earth — the differences among the three
sets of co-efficients are not excessive.
c
200
IBO
160
r40
120
100
60
60
40
20
10
TT-
toioV
1
1 lN2o.'ooo'^
//'
/
Y
kI:Oio;
/
1 IN 1,0
/
^^.^'-^
//
/
/
/
y
/
'■
/
/
1
/
/ h
/
\m 10
00 J
1 !
/
^
Ill
//
If
/
rpfinq)
1
/-
.- —
~
If
/_ ^
/
_ _
CoC-zsu)
1''
A
i
, -- '
y
If 1
///
^
^
r .'
Cm (-017)
/
f
///
X
TCkC-oit)
1
/,'U
/
^^
CflC-633) 1
/ f
/ 1
' ^
1 rN 1,000 J
//'
fj
/
1/^
/
/x
•{y
P
uy
^
1
/
^
y
Ck(-02251
1
^
,y
1
fy
■y
^■'''
1
/
/
y
^■-■^>'
CB(2-3a
(
//
//
^>-
"^ y^
/
V ,
^
y
f
\
-'^
^
/J
""
1 IN 20O0O3
'/
1 ,
y
/
X
^
'/
/
J
/
y
y
CKtOji)
,' 1
y
// /
/
x-
y
1 IN 1.000 J
/ ^
'/ /
/
" /
y
-r^-
Cm (-035)
1 ^
" ^
1.-'
•' /
/
//-
%'
/j
//
/ ^
' 1
' /
^
1 f
.<<
,y
/f
/^/
r
//
/
/
f /
/
3 -4 -7 I t-5 2 25 3 S-S 4 |
Fig. 113a.
N
194 HYDRAULICS
The empirical formulaa connecting the different values of the
co-eiRcients are as follows : —
Bazin's original co-efficients : i
V'('4)-
Kutter's co-efficients :
,, . , 1-811 , -00281
Bazin's new co-efficients :
157-6
C=-
The quantities a, ji, N and y are all constants depending on the
nature of the channel.
12. Rugosity Co-efficients. — The kinds of materials for which
various values of N have been generally accepted are as follows.
Unless otherwise stated all are supposed to be in good order and
joints smooth.
•009 Timber planed and perfectly continuous.
•010 Timber planed. Glazed and enamelled niateriak.
Cement and plaster.
•Oil Plaster and cement with one-third of sand.
Iron, coated or uncoated.
•012 Timber unplaned and perfectly continuous.
Concrete.
New brickwork (joints in perfect order).
•013 Unglazed stoneware and earthenware.
].''oul and slightly tuberculated iron.
Good brickwork and ashlar.
•015 Wooden frames covered with canvas.
Rough-faced brickwork. Well-dressed stonework.
•017 Fine gravel well rammed. Hubble iu cement.
Tuberculated iron.
Brickwork, stonework, and ashlar in inferior condition.
-020 Coarse gravel well ranuued.
C'oarse rubble laid dry. Rubble in inferior condition.
' Those are not now usuJ AU tables and diagrams show Bazin's new co-
elliciouts.
OPEN CHANNELS — UNIFORM FLOW 195
For earthen channels the following are the general values : —
•017 Channels in very good order.
•020 „ good order.
■0225 „ order above the average.
■025 „ average order.
•0275 „ order below the average.
•030 „ bad order.
•035 ,, very bad order.
A channel in very good order is free from irregularities, sharp bends,
lumps, hollows, snags, or other obstructions, weeds and overhanging
growth. A channel having all the above irregularities (or even a few
of them in excess) would be in very bad order. The above descrip-
tions are of course brief and general. The selection of the proper
value of iVin any particular case requires judgment and experience.
There are of course channels requiring values of N intermediate to
the above. The larger a stream the less its velocity is affected by
changes in roughness. In any description of a channel some idea
of its size should be conveyed. Overhanging growth which would
have little effect on a large stream may have great effect on a small
one. Small streams have — allowing for the difference in size —
sharper bends and greater irregularity of cross-section. There is
a tendency to underestimate N in such streams.
Regarding rivers and large canals some values of iT are given
in art. 9, but the figure •OlS refers to a brick channel. In other
rivers Kutter found iV" to vary from "025 to "042. In small torrents
— discharges of such are often observed to ascertain the run-off
of the rainfall — N may be •OS to '08 or even more. The roughness
of a channel is not necessarily the same at all parts of the bed
and sides. Therefore in any channel N may vary as I) varies.
In the Punjab canals N is generally taken to be •0225, but when
the channel has been worn very smooth and even, N has sometimes
been found to be as low as 016. In designing the large canals
of the Punjab Triple Canal Project, N was taken, by Sir John
Benton, to be ^020 for the Upper Jhelum Canal but ^0225 for the
Upper Chenab and Lower Bari Doab Canals, where it was expected
that more silt would be brought in, channels carrying much silt
being considered liable to have rougher beds than others.^ When
mud has deposited in a canal the channel may be very smooth. It
may be rough when sand deposits or when scour is going on.
For concrete pipes of 30 inches and 46' inches iVhas been found to
'^ Min. Proc. Inst. C.E., voL ooi.
196 HYDRAULICS
be '01 2. In the 14'5-foot concrete-lined tunnel recently constructed
for the New York water supply N was found to be -0124. For
very smooth concrete If has been found to be -Oil. Reinforced
concrete is now used for large pipes. The deposits which occur
in brick sewers may increase the roughness somewhat, but they
may fill up and make smooth any eroded mortar joints. Vitrified
stoneware in large sewers gives great smoothness as compared with
concrete, but this is in practice no advantage, because the distortion
of the pipes in burning causes irregularity at the joints.
The kinds of channels corresponding to Bazin's y are as
follows : —
■109 Cement, planed wood.
•290 Planks, bricks, out stone.
•833 Rubble masonry.
1^51 Earth if very regular, stone revetments.
2 35 Ordinary earth.
3^17 Exceptionally rough (beds covered with boulders, sides with
grass, etc.).
13. Remarks. — Besides the causes of discrepancies among the
values of C mentioned in chapter ii. (arts. 9 and 11) there are
others. On the Mississippi and Irrawaddy V was obtained by
the double float which gives erroneous results (chap. viii. art. 9).
The results of over a hundred discharges observed near the head
of a large canal in India, when atranged into groups according to
the depth of silt in the canal, show the average value of JV to be
•025 when there is little or no silt, but •OlS when the depth of silt
is from -5 foot upwards. Silt generally deposits in a wedge, the
depth being greatest near the head of the canal. It is therefore
probable that the want of uniformity of the flow gave a some-
what enhanced value to C, and consequently too low a value
to N. This would, however, account onlj- partially for the low
value of N, and it is probable that its correct value is not more
than •Oie in the silted channel. The above values are the average
ones. In individual discharges N varies enormously. For one
particular depth of silt it varies from -009 to '030. These varia-
tions may bo accounted for partly by real variations in the rough-
ness of the channel, which often becomes very irregular when
scouring is going on actively, partljr liy errors^ in the observations
of the individual surf aco-slo pes, and partly by \ariations in the
degree of the variability of the flo\\-.
For two channels equal as regards roughness of surface and
value of E, N is less when the profile of the section is semicircular
or curved than when it is angular. In Bazin's experiments on
' These may have been coniiderable (see ohap. viii. art, 2).
OPEN CHANNELS — UNIFORM FLOW 197
small channels C is 5 to 9 per cent, less for a rectangular section,
even though the depth was only J^ to ^ of the width, than for a
semicircular channel. The difference is probably due to the effect
of the eddies produced at the sides (art. 7). The co-efficients in the
tables may be taken to be for average sections, the section being
neither a segment of a circle nor a rectangle. (See also art. 4.)
In earthen channels N seems to be particularly low when the
ratio of width to depth is great. On the river Eavi at Sidhnai
the value of iV, deduced from a long series of observations, is often
•008 or '010, and never very much higher. The bed is often
silted, but not always. The flow is practically uniform, and the
slope observations were checked with a view to discovering any
error. The river is straight, very regular, about 800 feet wide,
and 6 feet to 10 feet deep. The case was specially investigated,
and it seems to be proved that iV" at this site is not above '010.
It is probable that the low value is due to the small effect of eddies
from the sides, as compared with narrower streams and to the
regularity of the flow. Generally streams as wide as the Eavi
are irregular. The river is straight for five miles upstream of the
discharge site and one mile downstream, a reach unique, perhaps,
among the rivers of the world, but its great length cannot be
the cause of the low value of iV. The silt is caused by a dam a
mile below the discharge site. In floods the dam is removed, and
the silt then scours out. Thus the bed is probably roughest for
the greatest depths of water. In spite of this, JV is very much
the same for all the depths from 6 feet to 10 feet, and C some-
where about 200.
Section V. — Movement of Solids by a Stream
14. Formulae and their Application. — The observations made
by Kennedy, and referred to in chap. ii. (art. 23), were made in
India on the Bari Doab Canal and its branches, the widths of the
channels varying from 8 feet to 91 feet, and the depths of water
from 2'3 feet to 7 '3 feet. The beds of these channels have, in the
course of years, adjusted themselves by silting or scouring, so that
there is a state of permanent rigime, each stream carrying its full
charge of silt. It was found that the relation between D and F in
any channel was nearly given by the equation
F=-84Z)-6* . . . (71)
Put in a general form, the equation is
F=cX)™ . . . . (72)
198 HYDEAULTOS
The theory advanced in the paper quoted is that the silt sup-
ported per square foot of bed is P^D whore Pj is the charge of silt,
and the force of the eddies as F-, so that Pji? is as V^. If the
solids consisted only of silt m would be perhaps ^, but there is also
rolled material. Tlie silt discharge is BD VF^, or is as B V^. The
rolled material is supposed to be as B V, and relatively small, and
the total solid discharge is thus as a function of V, varying less
rapidly than V^, say as F". On the Bari Doab Canal n was 2-56.
For, since D'^* is as F, I) is as Fi'^a, and BDVP as BPV^-^.
The equation
r=l-05Di . . . (73)
agrees nearly as closely as equation 71 with the observed results.
The equation
F= -gSi)-" . . . (73a)
has also been suggested. ^
All the above equatio.ns are partly empirical, and obviously apply
only when the silt and rolled material bear some sort of proportion
to each other. In theoretical equations of general application silt
and rolled material would have to be considered separately. If
there is silt alone, equation 72 may be of the true form for all cases,
m being probably ^ or less. If there is rolled material and no silt,
as in a clear stream rolling gravel or boulders, the moving force
depends on the bed velocity, J'j, and Z> will be absent from the
equation, or will enter into it only in so far as the ratio - * may
depend on D.
Regarding equation 71 as a semi-empirical working equation —
and no more has been claimed for it — applicable to canal systems
and streams carrying silt and fine sand, its practical importance is
very great. It is now known that in order to prevent, say, a
deposit in any reach or branch, /'must not be kept constant, but
be altered in the same manner as D. Whether it be altered as
D"* or i)4 does not, for moderate changes, make very much difier-
ence. The exact figures will in time be better known. In
designing a channel the proper relation of depth to velocity can
be arranged for, or, at least, one quantity or the other kept in the
ascendant; according as scouring or silting is the evil to be guarded
against.
The old idea was that an increase in T', even if accompanied by an
increase in 1), gave increased silt transporting power. In a stream
of shallow section this is probably correct, for F increases as D^,
' Procecdinii.i of I^imjab Eiujineering Congress, 1919.
OPEN CHANNELS — UNIFORM FLOW 199
that is, as fast as required by equation 71, and faster than required
by equation 73. In a stream of deep section a decrease in D gives
increased silt-transporting power. If the discharge is fixed, a change
in D or W must be met by a change of the opposite kind in the
other quantity. In this case widening or narrowing the channel may
be proper according to circumstances. In a deep section widening
will decrease the depth of water, and may also increase the velo-
city, and it will thus give increased scouring power. In a shallow
section narrowing will increase the velocity more than it increases
Di. In a medium section it is a matter of exact calculation to find
out whether widening or narrowing will improve matters.
If the water entering a canal has a higher silt-charge than can
be carried in the canal some of it must deposit. Suppose an
increased discharge to be run, and that this gives a higher silt-
carrying power and a smaller rate of deposit per cubic foot of
discharge, it does not follow that the deposit will be less because
the quantity of silt entering the canal is now greater than before.
Owing to want of knowledge regarding the proportion of rolled
material, and to want of exactness in the formulae, reliable cal-
culations regarding proportions deposited cannot be made.
Assuming equation 71 to be correct, Kennedy has determined
the following ' critical velocities,' or velocities below which silting
will occur in channels supplied with turbid water, such as that of
the Indian rivers, and has also published diagrams giving details.
i)=12 3 45 67 89 10
r=-84 1-30 1-70 2-04 2-35 2-64 2-92 3-18 3-43 3-67.
The preceding figures refer to heavy silt and fine sand, such
as enters canals taking off from the upper reaches of the Northern
Indian rivers. For reaches of the canals distant from their heads
or for canals taking off lower down the rivers, a velocity of '75 V to
■9 V may be substituted for V. For the fine sand of Sind, -84 in
equation 71 becomes -63, and for the coarse sand of the Cauvery
and Kistna rivers in Southern India 1-01. The proper figure
becomes known in each case from experience. Thrupp {Min. Proc.
Inst. C.E., vol. clxxi.) gives the following ranges of velocities as
those which will enable streams to carry different kinds of silt :—
i>=l-0 10-0
r= 1-5 to 2-3 3-5 to 4-5 Coarse sand.
F= -95 to 1-5 2-3 to 3-5 Heavy silt and fine sand.
F= -45 to -95 1-2 to 2-3 Fine silt.
In the chatmels on which Kennedy made his observations the
charge of silt was supposed to be equal in all cases. But actually
some of the coarser solids were gradually deposited, or drawn off
200 HYDRAULICS
by the irrigation distributaries — (small branches) — so that in the
lower reaches the silt charge was reduced. In these lower reaches
the lower values of D and F occur. If the silt charge had been
the same as in the upper reaches V would have been greater in
relation to'Z*. Therefore Kennedy's formula tends to show a some-
what too rapid decrease of V &a, D decreases. Possibly the index of
D in equation 73 or 73a is really more correct than in 71 and
in tha table of Kennedy's critical velocities given above, V, though
correct for a depth of about 7 feet, should perhaps be somewhat
higher than "84 for a depth of 1 foot.
The efiPect of a rising or falling stream on the movement of solids
is mentioned in chap. ix. art. 5. On the Irrawaddy it was found
that on the day of a high flood a great deepening of the channel
occurred at all the observation sites. ^ This may have been due
either to the rise or to the greater depth of water after the rise,
or to both. When a falling flood is accompanied by silting it may
be because water heavily charged with silt has entered the river
during the flood.
For special circumstances affecting silting or scour see chap. vii.
arts. 1, 2, 3, 7, 8, and 9.
The moving of rolled material must depend on V independently
of D. In a reach of the Sirhind Canal in Northern India the
rolled material formed, in a period of 20 days, 39 per cent, of the
whole.
It is probable that the force exerted by a stream on a solid which
it is rolling is more nearly as F^'^ than F^. This affects the above
mathematical investigation but not the practical results.^ It has
been seen that the exact form of the equation is not of extreme
importance.
Observations on circular sewers by Currall ' tend to show that in
order that road detritus and rubbish may be moved by rolling
or dragging, D must not be less than 2 5 inches in a 9-inch pipe
and 4'5 inches in a 27-inch pipe. For sewers a velocity of 2 to
3 feet per second is generally considered correct. The movement
or scour of solids, other than those in suspension, depends greatly
' Nott on the Irrawaddy River. Sarauelson (Government Press, Rangoon).
* If a number of bodies have similar shapes, and if D is the diameter of one
of tliem and V the velocity of the water relatively to it, the supporting or
rolling force is perhaps as K''' V- and the resisting force or weight as E^. If
these are just balanced D varies as F^', or the diameters of similarly shaped
bodies wliioh can just ho supported or rolled are as F'-' and their weights as
F° nearly.
" ilin. Proc. Inst. C.E., vol. cxcii.
OPEN CHANNELS — UNIFOKM FLOW 201
on how closely they are packed or stuck together, and the question
is outside the domain of Hydraulics.
One theory is that the power of a stream to transport solids
depends on the difference between the velocities of two adjacent
horizontal layers. Such layers of course do not slide on one
another but are eddying and intermixed. When V is below the
critical velocity V,, (chap. ii. art. 15) there are no eddies at all
and probably no sliding of one layer on another, the greater velocity
near the centre of the stream being accompanied by a general
deformation of the mass, as it might be in a column of india-rubber.
When V rises above Vp there are eddies everywhere and still no
sliding. There are general differences of velocity among the
horizontal layers. These differences are greater the rougher the
bed. So are the eddies caused at the bed. The theory just
mentioned does not seem to be practically different from the one
already considered.
The action of a stream on a vertical or very steep bank seems
to depend chiefly on V alone and not on the relation of l^ to D.
If V is less than 1 foot per second and the water is heavily silted a
deposit may occur on the bank, tending to narrow the channel.
This is especially likely to occur if there is vegetation on the bank.
If V is about 3 feet per second, scour of the bank is, with many
soils, likely to occur. This is independent of scour due to bends
(chap. vii. art. 1), and again is affected by vegetation.
Let it be required to design a channel to carry a given discharge
and to have a given relation of F to Z) so as to prevent silting
or scour. If S is not fixed there is an infinite number of such
channels. In deciding which to adopt the question of the actual
velocity comes in with reference to possible action on the bank.
Owing to these considerations and to general convenience it has
been found necessary in the Punjab to fix the approximate ratio
of W to D. Some of the figures are as follows : —
Q = 2 12 80 300 600 1100 2200 3000 c. ft. per second.
^=2 3 4 5 6 8 12 15
Let there be two channels, equal as to D and V but one having a
rougher bed than the other, and of course a steeper slope. The
bed velocity in the rougher channel will be the less but the
difference perhaps not very great, and in spite of it the strength
of the eddies formed at the bed will probably be greater in the
rougher channel. If a short length of channel is roughened the
local surface slope is increased, but, owing to the smallness of the
202 HYDRAULICS
length, D and V are not affected. A greater proportion of silt
is thrown up to the surface. This in no way affects Kennedy's
conclusions but is outside them. His channels did not vary much
in roughness.
15. Kemarks. — The channels in which the observations above
referred to were made have all, as stated, assumed nearly rect-
angular cross-sections, the sides having
become vertical (Fig. 114) by the deposit
on them of finer silt ; but the equations
probably apply approximately to any ^' ^
channel if B is the mean depth from ^^°- 'i*-
side to side, and V the mean velocity in the whole section.
If the ratio of V to D", say V to D"*, differs in different parts
of a cross-section, there is a tendency towards deposit in the parts
where the ratio is least, or to scour where it is greatest. There
is, of course, a tendency for the silt-charge to adjust itself to the
circumstances of each part of the stream, that is, to become less
where the above ratio is less, but the irregular movements of the
stream cause a transference of water transversely as well as
vertically, and this tends to equalise the silt-charge. In a channel
with not very steep side-slopes the angles at M, K (Fig. 115)
frequently silt up — -the velocity there being relatively low — and
the sides become steep or vertical. ^^ j,
Sometimes, even when the sides are
vertical, fine, silt adheres to them, and
the channel contracts, even though there
' ° Fig. 115.
may be no deposit in the bed. When
the bed is level across there frequently occurs a shoaling near the
sides, or a scour in the middle, and a marked
rounding-off at the lower angles. The section
thus tends to assume the form shown in
Fia no. ^ig- 116. When the bed is of sand, as in the
Bari Doab Canal channels, it remains nearly
level, because the sand at the sides rolls towards the centre.
It is clearly impossible to answer, in a general manner, questions
such as whether the embanking of a river, or confining it by
training-walls, will cause its bed to rise or to scour ; whether silt
will deposit on flooded land ; whether the minor arm of a stream
will tend to silt and become obliterated. Everything depends on
the charge of silt oiiginally carried, on the hardness of the
channels, and on the relations between D and /'.
OPEN CHANNELS — UNIFORM FLOW 203
When a channel is sandy the longitudinal section is often a
succession of small abrupt falls. After each fall there is a long
gentle upward slope till the next fall is reached. The sand is
rolled up the long slope and falls over the steep one. It soon
becomes buried. The positions of the falls of course keep moving
downstream. The height of a fall in a large channel is perhaps
6 inches or 1 foot, and the distance between the falls 20 to 30 feet.
A fall does not extend straight across the bed but zigzags, so
that the channel as viewed from above presents the appearance
of waves.
Some rivers in the northern hemisphere which flow in a southerly
direction have a tendency to shift their channels westwards. Tliis
is especially noticeable in some of the Indian rivers. The revolu-
tion of the earth has been ascribed as a cause. As the water ap-
proaches the equator its velocity of rotation about the earth's axis
increases. In latitude 30° a stream flowing south at 2 miles an
hour has its velocity of rotation increased in one hour from about
1300 feet per second to 1300'37 feet per second, or by "37 feet
per second. This is not a large amount in an hour, and the
pressure due to it must be a negligible quantity. Moreover, scour
depends on velocity not on pressure (chap. ii. art 23, cf. chap,
vii. art 1).
An irrigation branch channel, whether large or small, taking
off at a right angle from a canal, often receives more than its due
share of silt deposit. This is probably owing to the stirring up
of rolled material by the eddies formed at the oiF-take (cf. chap,
vii. art. 9). The off-take is a masonry 'head.' One of the
commonest remedies for silting is to make the floor of the head
higher than the bed of the canal — or to make the water pass over
a raised 'sill' or gate or both — so as to try to exclude rolled
material. But even in such cases the branch may silt. Fig. 116a
(section) shows the canal on the left. There are of course at the
off-take, transverse to the canal, velocity-of-approach currents some-
what as shown by the arrows. It has been stated in chap. ii.
art. 20 — also see chap. vii. art. 3 — that high velocity in the canal
reduces the discharge of a branch taking off from it at right angles,
and it has been argued that the branch draws in most of its water
from the lower half of the aperture because the water at that level
is moving relatively slowly. This consideration, however, has not
much force with the usual large apertures and moderate velocities.
It has also been argued that the cross currents (art. 7 and fig. 108)
cause silt to be carried towards the sides of the canal at a some-
204 HYDKAULICS
what low level, but it has been seen that the general effect of
the currents is to equalise the silt charge. In the absence of
currents the water near the sides would be less highly charged than
that in the centre.
An arrangement devised by King,i to reduce or prevent the
deposit of silt in a distributary, consists in the fixing of vanes on
the bed of the canal at AB in such a way as to throw oif the lower
water towards G. A compensating surface flow occurs from D to F.
The water in the canal is given a rotatory movement. This has
been tried with excellent results, the main channel, however, being a
distributary and the branch channel a ' water-course ' whose head
was only "5 foot square with no wing walls, and its floor level with
the distributary bed. It seems probable that the chief benefit is
due simply to the throwing oif of the silted water and its replace-
ment by clearer water. Simple roughening of the bed might not
be effectual. Another plan is to substitute for the vanes a low
masonry spur whose width is gradually reduced, in going upstream,
at the rate of 1 in 4 so that the spur throws off the lower water
of the canal. This has been tested with complete success, the
silting of a distributary — not merely a water-course — having been
cured. In all cases the bed of the main channel at the off-take
has to be pitched ; otherwise severe scour would be caused by
the disturbance.
It has been argued that a low velocity of inflow through MN
is desirable. Water flowing upwards from the bed of the canal will
be able to carry more sand the greater its velocity. Velocity of
approach, however, depends on the discharge of the aperture, and
this depends not only on the velocity at MN but on the depth
MN. A remedy for silting is no doubt the reduction of the depth
MN — the length of the aperture being increased to give the proper
value of Q — and the increase of the height NP.
The case of the head-works of a large canal is similar. The
water flows over a sill which can be further raised by gates. This
is independent of the formation of a 'silt trap' in the river by
a closure of the gates of the weir which runs across the river,
succeeded in due course by their reopening and the simultaneous
closure of the canal.
If a distributary has no raised sill or gate the flow of entry
is still like tliat over a submerged weir (chap iv. art. 15), and
the lower the velocity of approach the better. Opinion tends to
favoiir wide head openings for distributaries. Not only is velocity
' Proceedings of Punjab Engineering Conference, 1918.
OPEN CHANNELS— UNIFOEM FLOW
205
of approach reduced but eddies are reduced. They would be further
reduced by making the opening bell-mouthed (c/. chap. ii. art. 20).
Plan.
m t
.j:\.
Road.
Bed of Canal.
Elevation.
Fig. 116a.
Notes to Chapter VI.
Dependence of U on D on a Vertical in a Gross-Section (art. 8). —
" We have here a most remarkable section of a great river, in which
from one bank the bottom slopes in the same direction for a dis-
tance of over 3700 feet with the regularity of railway gradients,
the depths increasing from nil to 72 feet." The above refers to
the Parana. 1 Revy found that the surface velocity Ug varied as D,
Hydraulics of Great Rivers.
206 HYDEAULICS
and concluded that since i.= increases somewhat with D, U must
vary as Z)" where n is greater than unity. He found a similar
result on the Uruguay. But in each case he rejects an observation
— at or about the maximum depth — which if accepted would tend
to show that U, did not increase so rapidly as D. The rejections
were made on the ground that the depths at the points under
consideration were probably local, i.e. that they occurred only on
the single cross-section taken. There seems to be no proper
evidence as to this. The observations at the points considered
seem to have about as much weight as any of the others. Moreover,
on the Parana — this was the site of the most important experiments
— the observation site was at a bend of about 12 miles radius,
the width of the river being nearly a mile. The greatest depths
were of course near the concave bank (chap. vii. art. 1) and the
velocity would be somewhat greater than if the stream had been
straight, even with the same section. The total number of Revy's
observations was quite small. The flow appears to have been
steady. It seems clear, however, that U varied more nearly as D
than as D\ It cannot be supposed that the law governing two
wide portions of a stream is different from that governing two
separate wide streams, and these observations of Revy's afford some
evidence that C increases, with great depths, more than has been
supposed. He himself left thp matter to be explained by others.
Average Sections. — An earthen channel is seldom so regular that
any two parallel and neighbouring longitudinal or cross-sections
are exactly alike. In discussions such as those in the present
chapter average sections are always meant. A single section may,
as in the above case of the Parana, contain, for instance, a shallow
which is local, that is, does not extend to adjacent sections. At
such a shallow F, instead of being less than on neighbouring
verticals, is likely to be greater because of the rush of water
over it (c/. chap. vii. art. 2).
Examples
Explanation. — The explanation given under ex;unples in Chapter
V. applies also to open channels. If only one factor, say i^, is
fixed an infinite number of channels can be designed to carry a
given discharge, but usually other factois are determined by
practical considerations : the ratio of the side-slopes, say, by the
nature of the soil, and the ratio of JF to- D, say, by the velocity
OPEN CHANNELS — UNIFORM FLOM^
207
desirable or the solid-moving power required. If V must not
fall below a certain minimum this can be arranged by keeping li
large enough, or if this cannot be done, by altering S, N,
or Q. If V is not to exceed a certain maximum B can be kept
down, or S can be reduced to any extent by placing falls in the
channel.
Example 1. — Find the discharge of a stream with vertical sides
and 15 ft. wide when D = b-Q ft., iV=-017, and S=l in 5225.
From table xliii. A = 1b and ^i^=l-74-. From table xxxv.
C^i?=183. From table xxviii. a slope of -j^jVir gives F=2-59,
and the percentage to be deducted is f§# = 2-2, making F=2-53.
Then 0=75x2-53 = 189-8 c. ft. per second.
Example 2. — Design a channel with side-slopes 1 to 1 to dis-
charge 1000 c. ft. per second, S being -5-^Vo ^^d N= -0225. The
figures in the annexed statement show the results of successive
trials, the bed-width being 40 ft. It is clear that a depth of
7 '13 ft. gives the requisite discharge.
1st trial.
2nd trial.
Srd trial.
Bed-width, .....
40
40
40
Depth
7-5
7-25
7-0
ji from table xl v.,
356-3
342-6
329
^E from table xlv.,
2-41
2-38
2-34
Cv'^ from table XXX vii..
2^6
212
208
V from table xxviii. , .
3-05
3-00
2-94
Q=Ar, . .
10S7
1028
9C)7
1
Example 3. — In the preceding example let V be limited to
2-5 ft. per second. Find the minimum bed-width.
A must be 400. From table xxviii. C JP,, is 177, and this in
table xxxvii. gives ^ii=2-08. From table xlv., a bed-width of
80 ft. and depth 4'75 ft. gives practically the required values of
A and JR.
Example 4. — A channel 20 ft. wide with side-slopes ^ to 1 and
depth 5 ft. has to discharge 240 c. ft. per second, N being '025.
Find ,S'.
240
From table xliv. ^ = 112-5 and JH^l-'^Q. Then V=,
112-5
2-13 ft. per second. Assume tS" to be ^^^Vff. Then table xxviii.
gives C;^/ii=151, which corresponds in table xxxviii. to JB=2-0.
Therefore S has been assumed too low. Assume it to be jy^nj'
then CJR=U-2-8 and ^.^=1-92. To be exact ^.S* must be
5-0
4-5
4-0
3-5
30
2-35
2-20
2-04
1-87
1-70
208 HYDRAULICS
1"92
increased in the ratio r-^^-, or by 1 per cent, nearly, that is,
'S'=ti:ti-
Example 5. — Keeping Q the same, alter D and S in the last case
so as to give the necessary ratio of V to D to prevent silting
according to the rules of art. 14.
The statement given below shows that if D is reduced to 3-25 ft.
S will be as before (1 in 4410), but fF must be increased to 40 ft.
If ?F is left unaltered D can be 4-75, but S'must be increased to
about 1 in 3572. In a short channel, or one containing falls, it
would be easiest to increase S, but otherwise it would be necessary
to widen.
Depth of water,
Velocity according to above rule,
Mean width of channel to make\ ^^,~ .^ , .„ 90.. og.g ,'?
0=240 c. ft. per second, ./
Bed-width of channel to nearest"! ^^ 99 97 05 15
foot, /
^i? from table xliv., . . 1-87 1-85 1-79 1-73 1-64
C^-K from table xxxviii., . 137 135 129 123 114
5 (from table xxviii.) to give FjgggQ g^g^ ^qqq ^320 ^-qq
as above, 1 in . . J
Example 6. — In a channel A is found to be 48 sq. ft., JB is
14 ft., Q is known to be 100 c. ft. per second, and S is -^ysji-
Find C and N.
V is -yy-=2-08 ft. per second. From table xxviii., if 8=-^^^^,
C ^R=\IL An addition of 61 to 3000 decreases V by 1 per
cent., .■. an addition of 100 decreases F^by 1'6 per cent., and C JB
must be increased by 1'6 per cent., that is, it is 115-8. Then
C= =82'7, which (table xxxvi.) corresponds very nearly to
i\r=-020.
Example 7. — In a channel with vertical sides, 70 ft. wide and
5 ft. deep, the central surface velocity is 3 ft. per second, N is
•025. What is VI,
From the table on page 187 jS is •89. From the table on page
183 a is -945. Then r=3 x ^89 X •945 = 2^52 ft. per second.
OPEN CHANNELS— UNIFORM FLOW
209
Tables of Kutter's and Bazin's Co-efficients
These are given to three figures, and the engineer who uses
them will be fortunate if the actuals come out so as to agree with
the third figure or even come near it. To add a fourth figure is
useless, and it would render the tables bulky and less convenient.
The values of C JR have been obtained from the four-figure values
of C, and the figures in excess of three struck out.
As N increases the difference in C becomes less in proportion
to the change in N. Hence it is not necessary to give C for
i\^=-0325.
Table XXIX.
—Kutter's Co-efficients (iV=-009).
VR
1 in 20,000
1 in 15,000
1 in 10,000
1 ia 6,0C0
1 in 2,500
1 in 1,000
C
C^B
C
98-9
WR
C^R
C
CVR
C
C^/B
C
C^/R
•4
934
ZTi
39-5
106
42-2
114
45^6
119
47-7
123
49^1
•45
101
45^6
107
48
113
51-0
122
54-7
127
57
130
58^5
•5
108
54^2
114
56-8
120
60^1
128
64-2
133
666
137
68-2
•55
115
633
120
66-2
127
69-6
134
73-9
139
76^5
142
78-1
•6
121
72^7
126
75-8
132
79-4
140
84
145
86-7
147
8S^4
•65
127
82^5
132
85-8
138
89-5
145
94-2
149
97
152
98^8
•7
133
92^7
137
961
143
100
150
105,
154
108
156
109
•8
142
114
147
117
152
122
158
126
162
129
164
131
•9
151
136
155
140
160
144
165
149
168
151
170
153
1
159
159
163
163
167
167
171
171
174
174
175
175
M
166
183
169
186
173
190
177
194
179
197
180
198
1^2
173
207
175
210
178
214
181
218
183
220
184
221
1^3
178
232
180
235
183
238
185
241
187
243
188
244
1^4
184
257
185
259
187
262
189
265
190
267
191
267
1-5
188
283
190
285
191
287
193
289
193
290
194
291
1-6
193
309
194
310
195
311
196
313
196
314
197
314
VI
197
335
197
336
198
336
198
337
199
338
199
338
1-8
201
362
201
362
201
362
201
362
201
362
201
362
rg
204
388
204
388
204
387
203
386
203
386
203
386
2
208
415
207
414
206
413
205
411
205
410
205
409
2-1
211
443
210
440
209
438
207
436
207
434
206
433
22
214
470
212
467
211
464
209
460
208
459
208
457
2-3
216
497
215
494
213
490
211
485
210
483
209
481
2-4
219
525
217
520
215
516
213
510
211
507
211
506
2-5
221
553
219
547
217
541
214
535
213
532
212
529
2-6
223
581
221
574
218
568
215
560
214
556
213
554
2-1
226
609
223
601
220
594
217
585
>215
581
214
578
2^8
228
637
224
629
221
620
218
610
216
605
215
602
2-9
229
665
226
656
223
646
219
635
217
630
216
626
3
231
694
228
683
224
673
220
660
218
654
217
650
210
HYDRAULICS
Tabl-e XXX. — Kutter's Co-efficients (iV=-01).
VR
1 in 20,000
1 in 16,000
] in 10,000
1 in
6,000
1 in 2,500
1 in 1,000
C
CVR
C \ CVR
C
91-4
CVR
36-6
C
Cv'JJ
C
41-5
C
WR
42-7.
■4
81
32^4
85^7
34-3
99
396
104
107
•45
87-9
39-6
926
413
98^3
44-3
106
47-6
110
497
114
5ri;
•5
94^4
47-2
991
49^6
105
52^4
112
56
117
58-2
120
59-7,
•55
100
55-2
105
57-8
111
60-8
118
64-7
122
67
125
68-6
•6
106
63-6
111
66-3
116
69-6
123
73-7
127
761
130
77-7
•65
111
72^3
115
75-3
121
78-7
128
82-9
131
85-4
134
87
•7
116
%\-4
121
84-5
126
88
132
92-3
136
94-9
138
96-6
•8
126
100
130
104
134
107
140
112
143
114
145
116
•9
134
120
137
124
141
127
146
132
149
134
151
136
1
141
141
144
144
148
148
152
152
155
155
156
156
M
148
162
150
166
154
169
157
173
159
175
161
177
1-2
154
184
156
187
159
190
162
194
164
196
166
198
1^3
159
207
161
209
163
212
166
216
167
217
168
219
1-4
164
230
166
232
167
234
169
237
171
239
171
240
1-5
169
253
170
255
171
257
173
259
174
260
174
261
1-6
173
277
174
278
175
280
176
281
176
282
177
282
1-7
177
301
178
302
178
302
178
303
179
304
179
304
1-8
181
325
181
325
181
325
181
326
181
326
181
326
1-9'
184
350
184
349
184
349
183
348
183
348
183
347
2
187
375
187
373
186
372
185
370
185
370
185
369
21
190
400
189
398
188
395
187
393
187
392
186
391
2^2
193
425
19-2
422
191
419
189
416
188
414
188
413
2^3
196
450
194
447
193
443
191
438
190
436
189
435
2^4
198
476
196
471
194
466
192
461
191
458
190
457
2^5
201
501
198
496
196
490
194
484
192
481
192
479
2-6
203
527
200
521
198
514
195
507
194
503
193
501
2'7
205
553
202
546
199
538
196
530
195
526
194
523
2-8
207
579
204
571
201
562
198
553
196
548
195
545
29
209
605
206
596
202
586
199
576
197
571
196
567
3
210
631
207
621
204
611
200
599
198
593
196
589
OPEN CHANNELS — UNIFORM FLOW
211
Table XXXL— Kutter's Co-efficients (iV=-011).
a/B
1 in 20,000
1 in
6,000
lin
0,000
1 in 5,000
1 in 2,600
1 in 1,000
C
CV-E
C
CVR
CV-R
CVR
CVR
C
CVR
■4
7M
28^5
75-3
391
80^3
32-1
81-1
34-S
91-3
36-5
94^1
37-6
•45
77-4
349
81-6
36-8
86-6
39
931
42
97^5
43-8
100
451
■5
83-3
41^6
87-4
43^7
92^5
46-2
99
49-5
103
51^5
106
52^8
•55
88-8
48^8
92-9
51-1
97-9
53^8
104
57-3
108
59^5
111
60-8
■6
94
56-4
98
58^8
103
61-7
109
65^4
113
67-7
115
691
•65
98^9
64-2
103
66-8
108
69^8
113
73^7
117
76^1
119
77^6
•7
104
72-4
107
75^1
112
78-2
118
82-3
121
84^7
123
86^2
•8
112
89^5
115
923
120
956
125
99-8
128
102
130
104
•9
120
108
123
110
127
114
131
118
134
120
136
122
1
126
126
129
129
133
133
137
137
139
139
140
140
M
183
146
135
149
138
152
142
156
144
158
145
159
1-2
138
166
141
169
143
172
146
175
148
177
149
178
1-3
144
187
145
189
147
192
150
195
151
197
152
198
1-4
148
208
150
210
151
212
153
215
154
216
155
217
V5
153
229
154
231
155
233
156
235
157
236
158
237
1-6
157
251
158
252
158
253
159
255
160
256
160
256
1-7
161
273
161
274
162
275
162
275
162
276
162
276
rs
164
296
164
296
164
296
164
296
165
296
104
,296
1-9
168
318
167
318
167
319
167
317
167
316
166
316
2
171
341
170
340
169
339
169
337
168
337
168
336
2-1
174
364
173
363
172
361
171
358
170
357
170
356
2-2
176
388
175
385
174
382
172
379
172
378
171
376
2-3
179
411
177
408
176
404
174
400
173
398
172
397
24
181
435
179
431
178
426
176
421
175
419
174
417
2-5
184
459
182
454
179
448
177
442
176
439
175
437
2-6
186
483
183
477
181
471
178
464
177
460
176
458
2-7
188
507
185
500
183
493
180
485
178
481
177
478
2-8
190
531
187
523
184
515
181
506
179
502
178
498
2-9
191
551
188
547
185
537
182
528
180
522
179
519
3
193
580
190
570
187
560
183
549
181
543
180
539
212
HYDRAULICS
Table XXXII.— Bazin's and Kuttbr's Co-efficients.
VR
Bazin.
Kntter. N= -012
■y=-109
1 in 20,000
1 in 16,000
1 in 10,000
1 in 6,000
1 in 2,600
1 in 1,000
C
CVR
496
C
63^2
CVB
253
C
66-9
26-7
C
71-4
28-5
C
77-4
CVS
30-9
C
81^2
Cy/B
32^5
83 7
CVR
335
•4
124
•45
127
57-1
ii8^9
3M
72^6
326
771
34^7
83^137-4
86^9
391
89 2
403
•5
130
64-8
74^3
372
78
39
82-5
41-2
88-444^2
92^1
46
94-6
47^3
•55
132
72-6
79-3
43-7
83^1
45^6
87^5
48
9;f^3 51-3
96^9
53-2
99-2
54^5
•6
133
80
84-1
50^5
87-8
52^6
92^1
55-2
97^8
58-7
101
60-7
103
62
•65
135
87-7
88^6
57-6
92^2
59^9
96^4
62^6
102
■66^3
105
68-3
107
69^7
•7
136
95^4
929
65
96^2
67-5
101
70-3
106
74
109
76-1
111
77^6
•8
139
111
lOL
80^7
104
83^2
108
86^3
113
90^1
115
92-3
117
93^7
■9
141
126
108
97-2
111
99-8
114
103
119
107
121
109
123
110
»1
142
142
114
114
117
117
120
120
124
124
126
126
127
127
11
144
145
120
132
123
135
125
138
129
141
130
143
132
145
1-2
144
173
126
151
128
153
130
156
133
159
134
161
135
162
rs
145
189
131
170
132
172
134
175
137
177
138
179
139
180
1%
146
205
135
189
137
191
1.38
193
140
196
141
197
142
198
1-5
147
220
140
209
141
211
142
212
143
214
144
216
144
216
1-6
148
236
144
230
144
231
145
232
146
233
146
234
147
234
1-7
148
252
147
250
147
251
148
251
148
252
149
252
149
253
rs
149
267
151
271
151
271
151
271
151
271
151
271
151
271
1-9
149
283
154
292
153
292
153
291
153
290
153
290
153
290
2
149
299
157
314
156
312
156
311
155
310
155
309
154
303
21
150
301
160
335
159
334
158
331
157
329
156
328
156
327
2^2
150
330
162
357
161
354
160
352
159
349
158
347
157
346
2^3
150
346
165
379
163
376
162
372
160
368
159
366
159
365
2-4
151
362
167
401
165
397
164
393
162
388
161
385
160
384
2-5
151
378
169
423
167
418
165
413
163
408
162
405
161
403
2^6
151
393
171
446
169
440
167
434
164
427
163
424
162
421
2-7
151
408
173
468
171
462
168
455
166
447
164
443
163
440
2'8
152
424
175
491
173
483
170
476
167
467
165
462
164
460
2-9
152
440
177
514
174
505
171
497
168
487
166
482
165
479
3
152
456
179
536
176
527
173
518
169
507
167
501
166
498
31
152
472
32
152
487
33
152
503
3^4
153
519
Bazin's co-efficienta for higher values of iJR.
35
153
535
3-6
153
550
^/B = 5-0 7^0 8^0
37
153
566
0=154 155 155
3-8
153
582
3-9
153
598
4
153
614
OPEN CHANNELS — UNIFORM FLOW
213
Table XXXIIL— Bazin's and Kutter's Co-eeficients.
VR
Bazin.
Kutter. N= -013
T = -290
1 in 20,000
1 In 16,000
1 in 10,000
1 in 6,000
1 in 2,500
1 in 1,000
C
91-8
WR
36-7
C
56-7
C-/R
22-7
C
60
24
64
C^R
25-6
C
69-4
C^/R
27-8
C
72-8
CVR
29-1
C
75-2
CVR
301
•4
■5
99-9
50
36-9
33-5
70-3
35-1
74-3
37-2
79-7
39-8
83
41-5
85-3
42-6
•6
106
63-8
76-0
45-6
79-3
47-6
83'3
50
88-4
53
91-5
54-9
93-6
56-2
•7
115
78
S4-2
58-9
87-3
61
911
1)3-8
95-9
67-1
98-8
69-1
101
70-5
■8
116
92-6
91-6
73-3
94-6
75-7
98
78-4
102
81-9
105
84
107
85-4
•9
119
107
98-3
88-4
101
90-9
104
93-7
108
97-3
110
99-4
112
101
1
122
122
104
104
107
107
110
110
113
113
115
115
117
117
1-1
125
137
110
121
112
123
115
126
118
129
119
131
121
133
1-2
127
152
115
138
117
140
119
143
122
146
123
148
124
149
1-3
129
168
120
156
121
158
123
160
125
163
127
164
127
165
14
131
183
124
174
120
176
127
172
129
180
129
181
130
182
1-5
132
198
128
193
129
194
130
196
132
197
132
198
133
199
1-6
133
213
132
211
133
212
133
214
134
215
135
216
135
216
1-7
135
229
136
231
136
231
136
232
137
232
137
233
137
233
1-8
136
244
139
250
139
250
139
250
139
250
139
250
139
250
1-9
137
259
142
270
142
269
142
269
141
268
141
268
141
268
2
138
275
145
290
144
289
144
288
143
286
143
286
142
285
21
138
290
148
310
147
309
146
307
145
305
145
304
144
303
2-2
139
306
150
331
149
328
148
326
147
323
146
321
146
320
2-3
140
320
153
351
151
348
150
345
148
341
147
339
147
338
2-4
141
34 6
155
372
154
368
152
364
150
360
149
357
148
355
2-5
141
353
157
393
155
388
153
3S4
151
378
150
375
149
373
2-6
142
369
159
414
157
409
155
403
153
397
151
393
150
391
2-7
142
384
161
435
159
429
156
422
154
415
152
411
151
409
2-8
143
400
163
457
161
450
158
442
155
434
153
429
152
427
2-9
143
415
165
478
162
470
159
462
156
453
154
448
153
444
3
144
431
167
500
164
491
161
482
157
471
155
466
154
462
3-1
144
447
3-2
144
462
3-3
145
478
3-4
3-5
3-6
3-7
145
145
146
146
493
509
525
540
Hazin's co-efficients for higher values of ijli.
V-B= 4-5 5-0 6-0 7-0 8-0
0=1^8 149 150 151 152
3-8
146
556
3-9
147
572
4
147
588
214
HYDRAULICS
Table XXXIV. — Kutter's Co-efficients (7V=-015).
^n
1 in 20,000
1 in 16,000
1 in 10,000
1 in 5,000
1 in 2,600
1 in 1,000
C
C^R
C
CVfi
C
CVfi
C
wn
C
wn
C
1
•4
46-8
lS-7
49-4
19-8
h1-l
21-1
57-1
22-9
60
24
62
24-8
■5
55-5
27-8
58-3
29-2
61-6
30-8
66-1
33
68-9
34-4
70-8
35-4
■6
63'4
38
661
39-7
69-4
41-7
73-8
44-3
70-4
45-9
78-3
47-9
•7
70-4
49-3
73
51-2
76-2
63-4
80-3
56-2
82-8
58
846
59-2
■8
77-1
61-7
79-4
63-7
82-5
66
86-3
691
88-6
70-9
901
721
•9
83-1
74-8
85-4
76 8
88-1
79-3
91-5
82-4
9;{-6
84-2
94-9
85-4
1
88-6
88-6
90-6
90-6
931
931
961
96-1
97-9
97-9
991
991
11
93-6
103
9j-5
105
97-7
107
100
110
102
112
103
113
1-2
98-3
118
99-9
120
102
122
104
125
105
126
106
127
1-3
103
134
104
135
lOfi
137
107
140
lo9
141
109
142
1-4
107
149
108
150'
109
153
111
155
111
156
112
157
1-5
111
166
111
166
112
168
113
170.
114
171
114
172
1-6
114
182
115
183
115
184
116
185
116
186
117
187
1-7
117
199
118
200
118
201
118
201
119
202
119
202
1-8
120
217
120
217
120
217
121
217
121
217
121
217
1-9
123
234
123
234
123
233
123
233
122
233
122 232 1
2
126
252
126
251
125
250
125
249
124
248
124
248
21
129
270
128
269
127
267
126
2G5
126
264
125
263
2-2
131
288
130
286
129
284
128
281
127
2S0
127
279
2-3
133
307
132
304
131
301
129
298
129
296
128
294
2-4
136
326
134
322
133
318
131
314
130
312
129
310
2-5
138
344
136
340
134
336
132
331
131
328
130
326
2-6
140
363
138
358
136
353
134
347
132
342
131
342
2-7
142
382
140
377
137
371
135
364
133
360
133
358
2-8
143
402
141
395
139
388
136
381
134
376
133
374
29
145
421
M 3 ; 4 1 4
140
406
137
397
135
393
134
390
3
147
440
144 4;i2
141
424
138
414
130
409
135
405
OPEN CHANNELS — UNIFORM PLOW
215
Table XXXV. — Bazin's and Kutter's Co-efjficients.
V-B
Bazin.
Kutter. JV=-Oir
•)/=-S33
1 in 20,000
1 in 15,000
1 in 10,000
1 in 6,000
1 in 2,500
1 in 1,000
C
WR
C
C^R
c jcvJi
C
CV-B
C
48-2
GVR
19-3
C
50-5
WR
20-2
C
52-3
20-9
•4
51-1
20-4
396
15-9
41-8, 16-7
44-5
17-8
•5
591
29-6
47-2
23-6
49-524-7
52-3
26-1
56-1
28
58-3
29-2
60-1
30
•6
66-1
39-7
54-2
32-5
56-5I33-9
59-2
35-5
62-9
37-8
65-1
391
66-8
40-1
•7
71-9
50-3
60-5
42-4
G2-7
43-9
65-4
45-8
69
48-3
71
49-7
72-6
50-8
■8
77-1
61-7
66-4
531
68-5
54-8
71-1
56-9
74-3
59-4
76-2
60-9
Trl
62-2
■9
81-7
73-5
71-8
64-6
73-7
06-4
76-1
68-5
79-1
71-2
80-7
72-7
82-1
73-9
1
85-9
85-9
76-7
70-7
78-6
78-6
80-7
80-7
83-3
83-3
84-8
84-8
86
86
11
S9-6
98-6
81-4
89-5
83
91-2
84-8
93-3
87-2
95-9
88-5
97-3
89-5
98-4
1-2
93-1
1)2
85-7
103
87-1
104
88-7
106
90-7
109
91-7
no
92-6
111
1-3
961
126
89-7
117
90-8
118
92-2
120
93-9
122
94-7
123
95-5
124
1-4
98-8
138
93-4
131
94-4
132
95-4
134
96-8
136
97-4
136
98-1
137
1-5
101
152
96-9
145
97-6
146
98-5
148
99-4
149
99-9
150
100
151
1-6
104
166
100
160
101
IKI
101
162
102
163
102
164
103
164
1-7
106
180
103
176
104
176
104
177
104
177
104
177
105
178
1-8
108
194
106
191
106
191
106
191
106
191
106
191
106
191
1-9
110
208
109
207
109
207
109
206
108
206
108
205
108
205
2
111
223
112
223
111
222
111
221
110
220
110
219
110
219
21
113
237
114
240
113
238
113
237
112
235
111
234
111
233
2-2
114
251
116
256
116
254
115
252
113
250
113
248
112
247
2-3
J 16
266
119
273
118
270
116
268
115
264
114
262
114
261
2-4
U7
281
121
290
120
287
118
283
116
279
115
277
115
276
2-5
118
296
123
307
121
303
120
299
118
294
117
291
116
290
2-6
119
310
125
32-1
123
320
121
315
119
309
118
306
117
304
2-7
120
325
127
341
125
336
123
331
120
324
119
321
118
319
2-8
121
340
128
359
126
353
124
347
121
339
120
335
119
333
2-9
122
355
130
377
128
370
125
363
122
355
121
350
120
Zih
.3
123
370
132
394
129
387
126
379
123
370
122
365
121
362
3-1
124
385
133
412
130
404
128
395
124
385
122
380
122
377
3-2
125
400
135
430
132
421
129
412
125
401
123
394
122
391
3-3
126
415
136
448
133
439
130
428
126
416
124
409
123
406
3-4
127
430
137
467
134
456
131
444
127
431
125
424
124
420
3-5
127
446
139
485
135
473
132
461
128
447
126
439
124
435
3-6
128
460
140
503
136
491
133
477
129
463
126
454
125
450
3-7
129
476
141
522
137
508
134
494
129
478
127
469
126
464
3-8
129
491
142
540
138
526
134
510
130
493
127
484
126
479
3-9
130
506
143
559
139
544
135
527
131
509
12S
499
127
494
4
130
521
144
577
140
561
136
544
131
525
129
514
127
508
Bazin's oo-effioianta for higher values
of i?|
4-6g
'135
138
7
141
143,
216
HYDRAULICS
Table XXXVL— Kutter's Co-efficients (iV=-020).
VJ!
1 in 20,000
1 in 15,000
1 in 10,000
1 in 5,000
lin
2,500
1 in 1,000
C
CVR
13-4
C
35-7
Cv'JJ
C
Ov'Il
a
CVS.
C
CVS.
•4
32
12-8
33-6
14'3
38-7
15-5
40-6
16-2
41-9
16-8
■5
38-3
19-2
40-2
20-1
42-3
21-2
45-3
22-7
47-3
23-6
48-6
24-3
•6
44-2
26-5
46
27-6
48-2
28-9
51-2
30-7
531
31-9
54-4
32-6
■7
49-6
34-7
61-5
36
53-6
37-5
56-4
39-5
58-2
40-8
59-5
41-6
•8
54-7
43-8
56-3
45-1
58-4
46-7
611
48-9
62-8
50-3
64
51-2
•9
59-4
53-4
60-9
54-9
62-9
56-6
65-4
58-8
66-9
60-2
68
61-2
1
63-7
63-7
65-2
65-2
66-9
66-9
69-2
69-2
70-6
70-6
71-5
71-5
11
67-8
74-6
691
76
70-7
77-7
72-7
79-9
73-8
81-2
74-7
82-3
1-2
71-6
85-9
72-8
87-4
74-1
89
75-8
91
76-9
92-2
77-6
931
1-3
75-2
97-7
76-2
99
77-3
101
78-8
102
79-6
104
80-2
104
1-4
78-6
110
79-4
111
80-3
112
81-4
114
821
115
82-6
116
1-5
81-7
123
82-3
123
83-1
125
83-9
126
84-4
127
84-8
127
1-6
84-8
136
85-2
136
85-6
137
86-2
138
86-6
139
86-8
139
1-7
87-6
149
87-8
149
88
150
88-4
150
88-5
151
88-6
151
1-8
90-2
162
90-3
163
90-3
163
90-3
163
90-3
163
90-4
163
1-9
92-8
176
92-7
176
92-4
176
92-2
175
92
175
92
175
2
95-2
190
94-8
190
94-4
169
93-9
188
93-6
187
93-5
1S7
2-1
97-5
205
97
204
96-3
202
95-6
201
951
200
94-8
199
2-2
99-7
219
99
218
98-1
216
971
■214
96-6
212
96 1
211
2-3
102
234
101
232
99-8
230
98-5
227
97-8
225
97-4
224
2-4
104
249
103
246
101
243
99-9
240
99
238
98-5
236
2-5
106
264
104
261
103
257
101
253
100
251
99-6
249
2-6
108
280
106
276
104
271
102
266
101
263
101
262
2-7
109
295
108
290
106
285.
104
5:80
102
276
102
274
2-8
111
310
109
305
107
300
105
293
103
289
102
2S7
2-9
112
326
110
320
108
314
106
306
104
302
103
300
3
114
342
112
336
109
328
107
320
105
315
104
312
31
116
358
113
351
HI
343
108
334
106
328
105
325
3-2
117
374
114
366
112
357
109
347
107
342
106
338
3-3
118
390
116
382
113
372
109
361
110
355
106
351
3-4
120
407
117
397
114
386
110
375
108
368
107
364
3-5
121
423
118
413
115
401
111
388
109
3S1
108
377
3-6
122
439
119
428
116
416
112
402
110
394
108
390
3-7
123
456
120
444
116
431
112
416
110
408
109
403
3-8
124
472
121
4fi0
117
445
113
430
111
421
109
416
3 '9
125
489
122
475
118
460
114
444
111
435
110
429
4 127
506
123
491
119
475
114
458
112
448
110
442
OPEN CHANNELS — UNIFORM FLOW
217
Table XXXVII. — Bazin's and Kutter's Co-efficients.
Kutter. W= -0226
Bazin.
y = l-54
•4
1 in 20,000
1 in 16,000
1 in 10,000
1 in 5,000
1 in 2,600
1 in 1,000
C
32-8
CVS
131
C
27-4
CVR
11
C
CVR
C
CVB
C
CV-fi
C
CV-R
C
CVS
28-8
11-5
30-5
12-2
33
13-2
34-6
13-8
35-5
14-2
•5
38-6
19-3
33
16-5
34-5
17-3
36-3
18-2
38-8
19-4
40-5
20-3
41-5
20-7
•6
44-2
26-5
38-2
22-9
39-7
23-8
41-6
25
44-1
26-5
45-7
27-4
46-7
28
■7
49-2
34-4
43
30-1
44-5
31-2
46-4
32-5
48-9
34-2
50-4
35-3
51-3
35-9
•8
53-9
43-1
47-5
38
49
391
50-7
40-6
53-1
42-5
54-5
43-6
55-4
44-3
•9
58
62-2
51-7
46-5
53-1
47-8
54-8
49-3
56-*
51-2
58-3
52-5
59
53 1
1
61-9
61-9
55-7
55-7
57
57
58-5
58-5
60-5
60-5
61-7
61-7
62-3
62-3
11
65-7
72-3
59-4
65-3
60-5
66-6
61-9
68-1
63-7
70-1
64-7
71-2
65-3
71-8
1-2
69
82-8
63
75-6
63-9
76-7
65-1
78-1
66-6
79-9
67-5
81
68
81-6
1-3
72-1
93-7
66-2
86-1
67-1
87-2
68-1
88-5
69-3
90-1
70-1
91-1
70-5
91-7
1-4
74-9
105
69-3
97
70
98
70-8
99-1
71-9
101
72-5
102
72-8
102
1-5
77-6
116
72-3
109
72-8
109
73-4
110
74-2
111
74-7
112
74-9
112
1-6
80-3
129
75-1
120
75-5
121
75-9
121
76-4
122
76-7
123
76-8
123
1-7
82-6
140
77-7
132
77-9
133
78-1
133
78-4
133
78-6
134
78-6
134
1-8
84-9
153
80-2
144
80-2
144
80-3
145
80-3
145
80-3
145
80-3
145
1-9
86-9
165
82-6
157
82-5
157
82-3
156
82-1
156
81-9
156
81-9
156
2
88-9
178
84-9
170
84-5
169
84-2
168
83-7
167
83-5
167
83-3
167
2-1
90-9
191
87-1
183
86-6
152
86
181
85-3
179
84-9
178
84-7
172
2-2
92-5
204
89-1
196
88-5
195
87-7
193
86-8
191
86-2
190
85-9
189
2-3
94-3
217
91-1
210
90-2
208
89-3
205
88-1
203
87-5
201
87-1
200
2-4
95-8
230
93
223
92
221
90-8
218
89-5
215
88-7
213
88-3
212
2-5
97-4
244
94-8
237
93-7
234
92-3
231
90-7
227
89-8
225
89-3
223
2-6
99
257
96-6
251
95-2
248
93-7
244
91-9
239
90-9
236
90-3
235
2-7
100
271
98-2
265
96-7
261
95
257
93
251
91-9
248
91-3
247
2-8
102
284
99-8
279
98-2
275
96-3
270
94-1
264
92-8
260
92-2
258
2-9
103
298
101
294
99-6
289
97-5
283
951
276
93-7
272
93
270
3
104
312
103
308
101
303
98-6
296
96-1
288
94-6
284
93-8
281
31
105
326
104
323
102
317
99-7
309
97
301
95-4
296
94-6
293
3-2
106
340
106
338
103
331
101
323
97-9
313
96-2
308
95-4
305
3-3
107
354
107
353
105
345
102
336
98-7
326
97
320
96-1
317
3-4
109
369
108
368
106
359
103
350
99-5
338
97-7
332
96-7
329
3-5
110
383
110
383
107
374
104
363
100
351
98-4
344
97-4
341
3-6
110
397
HI
398
108
388
105
377
101
364
99
356
98
353
3-7
111
411
112
414
109
403
106
390
102
376
99-6
369
98-6
365
3-8
112
426
113
429
110
417
106
404
102
389
100
381
991
377
3-9
113
440
114
445
111
432
107
418
103
402
101
393
99-7
389
4
114
455
115
460
112
446
108
432
104
415
101
406
100
401
Bazin's co-efficients for higher values of i? | q..i^ j21 125
/Ji? = 4-5
7 8
129 132.
218
HYDRAULICS
Table XXXVIII. —Bazin's and Kutter's Co-EFriciENTS.
■
Kutter. «'=-025
Bazin.
Y = 2-35
VR
1 in 20,000
1 in 15,000
1 in 10,000
1 in 5,000
1 in 2,500
1 in 1,000
C
C^R
OVR
C
WB
C
WR
10-6
C
CVR
C
30
12
C
30-9
WR
12-4
•4
23-2
92-8
23-9
9-6
25
9-95
20-5
28-6
11-4
■5
27-6
13-8
28-9
14-4
30-2
15
31-7
15-9
33-9
17
35-3
17-7
36-3
18-2
•6
32
19-2
33-5
20-1
34-9
20-8
36-4
21-8
38-6
23-2
40
24
41
24-6
■7
36
25-2
37-9
26-5
39-2
27-4
40-7
28-5
42-9
30
44-2
30-9
45-2
31-6
■8
40
32
42
33-6
43-2
34-6
44-7
35-8
46-8
37-4
48
38-4
48-9
391
■9
43-6
39-2
45-8
41-2
46-9
42-3
48-4
43-6
50-3
45-3
51-5
46-4
52-3
47-1
1
47-1
47-1
49-4
49-4
50-5
50-5
51-8
51-8
53-6
53-6
54-6
54-6
55-4
55-4
M
50-2
55-2
52-8
58-1
53-8
59-2
55
60-5
56-6
6iJ-3
57-5
63-3
58-2
64
1-2
53-2
63-8
56
67-3
57
68-3
58
69-6
59-3
71-2
60-1
72-1
60-7
72-8
1-3
56
72-8
59-1
76-8
59-8
77-8
60-7
78-9
61-9
80-5
62-6
81-4
63
81-9
1-4
58-8
82-3
62
86-8
62-6
87-7
63-3
88-6
64-3
90
64-8
90-7
65-2
91-3
1-5
61-3
92
64-7
97-1
65-2
97-8
65-8
98-7
66-5
99-8
66-9
100
67-2
101
1-6
63-8
102
67-3
108
67-7
108
68
109
68
110
68-8
110
69
110
1-7
66-1
112
69-8
119
70
119
70-2
119
70-4
120
70-6
120
70-7
120
1-8
68-3
123
72-2
130
72-2
130
72-2
130
72-3
130
72-3
130
72-3
130
1-9
70-3
134
74-4
141
74-3
141
74-1
141
73-9
140
73-8
140
73-8
140
2
72-2
144
76-6
153
76-3
153
76
152
75-6
151
75-3
151
751
150
2-1
74-2
156
78-7
165
78-2
164
77-7
163
77-1
102
76-7
161
76-4
160
2-2
76-1
167
80-6
177
80-0
176
79-3
175
78-5
173
78
172
77-7
171
2-3
77-9
179
82-5
190
81-8
188
80-9
186
79-8
184
79-2
182
78-8
181
2-4
79-6
191
84-3
202
83-4
200
82-4
198
81-1
195
80-4
193
79-9
192
2-5
81-2
203
86-1
215
85-0
213
83-8
210
82-3
206
81-5
204
80-9
202
2-6
82-8
215
87-7
228
86-5
225
85-1
221
83-5
217
82-5
215
81-9
213
2-7
84-2
227
89-3
241
87-9
238
86-4
233
84-5
228
83-5
226
82-8
224
2-8
85-6
240
90-9
255
89-3
250
87-6
245
85-6
240
84-4
236
83-7
234
2-9
86-9
252
92-4
268
90-7
263
88-8
256
86-6
251
85-3
247
84-5
245
3
88-1
264
93-8
281
92
276
89-9
270
87-5
263
86-2
259
85-3
256
31
89-4
277
95-2
295
93-2
289
91
2S2
88-4
274
87
270
86
267
3-2
90-7
290
96-5
309
94-4
302
92
294
89-3
286
87-7
281
86-7
277
3-3
91-9
303
97-8
323
95-6
315
93
307
901
297
88-5
292
87-4
288
3-4
931
317
99
337
96-7
329
94
320
90-9
309
89-2
303
88-1
300
3-5
94-2
330
10)
351
97-7
342
94-9
332
91-7
321
89-9
315
88-7
311
3-6
95-3
343
101
365
98-7
355
95-8
346
92-4
333
90-5
326
89-3
322
3-7
96-2
366
103
379
99-7
389
96-6
358
93 1
344
91-1
337
89-9
333
3-8
97-2
369
104
394
101
383
97-5
370
93 8
356
917
349
90-4
344
3-9
98-2
383
105
408
102
396
98-2
383
94-4
368
92-3
360
91
355
4
99-2
397
106
423
103
410
99
396
95
380
92-8
371
91-5
366
Bazin's oo-effioienta for higher values of nf. '^p~ ,"^,'5
5
107
6
113
7
118
8
122.
OPEN CHANNELS — UNIFORM FLOW
219
Table XXXIX. — Kutter's Co-eeficients (iV=:-0275).
VJS
1 in 20,000
1 in 15,000
1 in 10,000
1 in 5,000
1 in 2,500
1 in IjOOO
C
CVR
C
CV-B
C
CVB
a
Cv'iJ
C
CVR
C
C/R
■4
21-2
8-5
22-2
8-9
23-4
9-4
23-2
101
26-4
10-5
27-2
10-9
•5
25-6
12-8
26-7
13-3
28
14
29-9
15
31-2
15-6
32
16
•6
29-8'
17-9
30-9
18-5
32-3
19-4
34-2
20-5
35-5
21-3
36-3
21 8
■7
33-8
23-7
34-9
24-5
36-3
25-4
38-1
26-7
39-3
27-5
40-2
28-i
■8
37-5
30
38-6
30-9
39-9
31-9
41-7
33-4
42-8
34-3
43-6
34-9
•9
41
36-9
42-1
37-9
43-2
39
45
40-5
46
41-4
46-8
42-1
1
44-4
44-4
45-3
45-3
46-5
46-5
48
48
49
49
49-6
49-6
11
47-5
52-2
48-4
53-2
49-4
54-4
50-8
55-9
51-7
56-8
52-3
57'0
1-2
50-5
60-6
51-3
61-5
52-2
62-6
53-4
641
541
64-9
54-6
65-6
1-3
53-3
69-3
54
70-2
54-8
71-2
55-8
72-5
56-4
73-4
56-9
73-9
1-4
56
78-4
56-6
79-2
57-2
80-1
58
81-2
58-6
82
58-9
82-4
1-5
58-6
87-9
59
88-5
59-5
89-3
60-1
90-2
60-5
90-8
60 '8
91-2
1-6
61
97-7
61-4
98-1
61-7
98-7
62-1
99-4
62-4
99-8
62-5
100
1-7
63-4
108
63-5
108
63-7
108
63-9
109
64-1
109
64-2
109
1-8
65-6
118
65-6
118
65-6
118
65-7
118
65-7
118
65-7
118
1-9
67-8
129
67-6
128
67-5
128
67-3
128
67-2
128
67-1
128
2
69-8
140
69-5
136
69-2
138
68-8
138
68-6
137
68-5
137
2-1
71-7
151
71-3
150
70-9
149
70-3
148
69-9
147
69-7
146
2-2
73-6
162
73-1
161
72-4
159
71-6
158
71-1
157
70-9
156
2-3
75-4
174
74-8
172
73-9
170
72-9
168
72-4
167
72
166
2-4
77-2
185
76-3
183
75-4
181
74-2
178
73-6
177
73-1
175
2-5
78-8
197
77-8
195
76-7
192
75-4
188
74-6
187
74-1
185
2-6
80-4
209
79-3
206
78
203
76-5
199
75-6
197
75
195
27
82
221
80-8
218
79-3
214
77-5
209
76-6
207
75-9
205
2-8
83-5
234
82
230
80-5
2-25
78-6
220
77-6
217
76-8
215
2-9
84-9
246
83-4
242
81-6
237
79-5
231
78-4
227
77-6
223
3
86-3
259
84-6
254
82-7
248
80-4
241
79-2
238
78-4
235
31
87-6
272
85-8
266
83-8
260
81-3
252
80
248
791
245
3-2
88-9
285
86-9
278
84-8
271
82-2
263
80-7
258
79-8
255
3-3
90-2
298
88-1
291
85-8
283
83
274
81-5
269
80-5
266
3-4
91-4
311
89-2
303
86-7
295
83-8
285
82-2
279
811
276
3-5
92-5
324
90-2
316
87-6
307
84-5
296
82-8
290
81-8
286
3-6
93-7
337
91-2
328
88-5
319
85-3
307
83-5
301
82-3
296
3-7
94-8
351
92-2
341
89-3
330
85-9
318
84-1
311
82-9
307
3-8
95-8
364
931
354
901
342
86-6
329
84-7
322
83-4
317
3-9
96-9
378
94
367
90-9
355
87-3
340
85-2
333
84
328
4
97-9
392
94-9
380
91-6
367
87-9
352
85-8
343
84-5
338
220
HYDRAULICS
Table XL. — Bazin's and Kutter's Co-efficients.
Kutter. N=-OiO
Bazin.
Y = 3-17
■4
1 in 20,000
1 in 15,000
1 in 10,000
1 in 5,000
1 in 2,500
1 in 1,000
C
17-9
C^Ii
C^/fi
C
WR
C
Cx/R
C
Cx/R
C
C^R
C
OVR
71-6
19
7-6
19-8
7-9
20-9
8-4
22-4
9
23-5
9-4
24-2
9-7
■5
21-5
10-8
23
11-5
24
12
25-1
12-6
26-7
13-4
27-8
•13-9
28-6
14-3
•6
25-2
15-1
26-8
161
27-8
16-7
29
17-4
30-7
18-4
31-8
19-1
32-5
19-5
•7
28-6
20
30-5
21-3
31-4
22
32-6
22-8
34-3
24
35-3
24-7
361
25-3
•8
31-9
25-5
33-9
27-1
34-8
27-8
36
28-8
37-6
301
38-6
30-9
39-3
31-4
■9
34-9
31-4
37-1
33-4
381
34-3
391
35-2
40-6
36-5
41-6
37-4
42-2
38
1
37-8
37-8
40-2
40-2
41-1
41-1
42-1
42-1
43-5
43-5
44-3
44-3
44-9
44-9
11
40-6
44-7
431
47-4
43-9
48-3
44-8
49-3
461
50-7
46-8
51-5
47-4
52-1
1-2
43-3
52
45-9
55-1
46-6
56
47-4
56-9
48-5
58-2
49-2
59
49-7
59-6
1-3
45-8
59-5
48-6
631
491
64
49-9
64-9
50-8
66
51-4
66-8
51-8
67-3
1-4
48-2
67-5
51-1
71-5
51-5
72-2
52-2
731
52-9
741
53-4
74-8
53-7
75-2
1-5
50-5
75-8
53-5
80-3
53-9
80-8
54-3
81-5
54-9
82-4
55-2
82-8
55-5
83-3
1-6
52-8
84-5
55-8
89-3
56-1
89-7
56-4
90-2
56-8
90-9
57
91-2
57-2
91-5
1-7
55
145
58
98-6
581
98-8
58-3
99-1
58-5
99-5
58-7
99-8
58-7
99-8
1-8
57
103
60-1
108
60-2
108
60-2
109
60-2
108
60-2
108
60-2
108
1-9
59
112
62 -2
118
621
118
61-9
118
61-8
117
61-6
117
61-6
117
2
61
122
64-1
128
63-9
128
63-6
127
63-2
126
63
126
62-9
126
21
62-8
132
66
139
65-6
138
65-2
137
64-6
136
64-3
135
64-1
135
2-2
64-6
142
67-8
149
67-3
149
66-7
147
66
145
65-5
144
65-3
144
2-3
66-3
152
69-5
160
68 9
158
68-1
157
67-2
155
66-7
153
66-3
153
2-4
67-9
163
71-2
171
70-4
169
69-5
167
68-4
164
67-8
163
67-4
162
2-5
69-5
174
72-8
182
71-8
180
70-8
177
69-6
174
68-8
172
68-3
171
2-6
71-1
185
74-3
193
73-4
191
72-1
188
70-7
184
69-8
182
69-3
180
2-7
72-5
196
75-7
204
74-7
202
73-3
198
71-7
194
70-8
191
70-2
190
2-8
73-8
207
77-2
216
76
213
74-5
209
72-7
204
71-7
201
71
199
2-9
75-2
218
78-6
228
77-2
224
75-6
219
73-6
213
72-5
210
71-8
208
3
76-5
230
80
240
78-5
235
76-6
230
74-5
224
73-3
220
72-6
218
31
77-9
241
81-3
2,i2
79-6
247
77-7
241
75-4
234
74-1
230
73-3
227
3-2
79-2
253
82-5
264
80-7
258
78-7
2;>2
76-2
244
74-9
240
74
237
3-3
80-3
265
83-7
276
81-8
270
79-6
263
77
234
75-6
250
74-6
246
3-4
81-5
277
84-9
289
82-8
282
80-5
274
77-8
265
76-3
259
75-3
256
3 '5
82 '6
289
86
301
83-9
294
81-4
285
78-6
275
76-9
269
75-9
266
3-6
83-8
302
87-1
314
84-9
307
82-3
296
79-3
285
77-6
279
76-5
275
3-7
84-7
313
88-2
326
85-8
317
83-1
308
79-9
296
78-2
289
77
285
3-8
85 -S
326
89-3
339
86-7
330
83-9
319
80-6
306
78-8
299
77-6
295
3 9
86-8
339
00-3
352
87 '6
342
84-7
330
81-2
317
79-3
309
78-1
305
4
87-8
351
91-2
365
88-5
354
85-4
342
81-9
327
79-9
320
78-6
314
Bazin's oo-officionts for higher values of III q— pi
92 97
6
103
7
108
113.
OPEN CHANNELS — UNIFORM FLOW
221
Table XLI. — Kutter's Co-efficients (^=-035).
VS
1 in 20,000
1 in 15,000
1 in 10,000
1 in 5,000
1 in 2,500
1 in 1,000
C
CVR
C
CVJi
C
cvii
C
CVS
C
CV-K
C
C^/lt
•i
15-6
6-3
16-3
a -5
17-1
6-84
18-3
7-3
19-1
7-6
19-7
7-9
■5
19
9-5
19-8
9-9
20-7
10-4
21-9
11
22-8
11-4
23-4
11-7
■6
22-3
13-4
13-1
13-8
24
14-4
25-3
15-2
26-2
15-7
26-8
16-1
•7
25-4
17-8
26-2
18-3
27-1
19
28-3
19-8
29-2
20-4
29-9
20-9
•8
28-3
22-7
29 1
23-2
30
24
31-3
25
32-1
25-7
32-7
26-2
■9
31-1
28
31-8
28-6
32-7
29-4
33-9
30-5
34-7
31-2
35-3
31-8
1
33-8
33-8
34-4
34-4
35-3
35-3
36-4
36-4
37-1
37-1
37-6
37-6
1-1
36-4
40
37
40-7
37-7
41-5
38-8
42-7
39-4
43-3
39-8
43-8
1-2
38-8
46-6
39-4
47-2
40
48
40-9
491
41-5
49-8
41-9
50-3
1-3
41 1
53-5
41-6
54-1
42-2
54-9
43
55-9
43-5
56-6
43-8
56-9
1-4
43-4
60-8
43-8
61-3
44-3
62
44-9
62-9
45-3
63-4
45-6
63-8
1-5
45-6
68-3
45-9
68-8
46-2
69-3
46-7
70-1
47
70-5
47-2
70-8
1-6
47-6
76-2
47-8
76-6
48-1
77
48-4
77-4
48-6
77-8
48-8
78a
1-7
49 '6
84-3
49-7
84-5
49-9
84-8
50-1
85-2
50-1
85-2
50-2
85-3
1-8
51-5
92-7
51-6
92-8
51-6
92-9
51-6
92-9
51-6
92-9
51-6
92-9
1-9
53-4
101
53-3
101
53-2
101
53
101
52-9
101
52-9
101
2
55-1
110
55
110
54-7
109
54-4
109
54-2
108
54-1
108
2-1
56-9
119
56-5
119
56-2
118
55-7
117
55-4
116
55-2
116
2-2
58-5
129
58-1
128
57-6
127
57
125
56-6
125
56-3
124
2-3
60-1
138
59-6
137
58-9
136
58-2
134
57-7
133
57-4
132
2-4
61-6
148
61
146
60-2
145
59-3
142
58-7
141
58-4
140
2-5
63-1
158
62-4
156
61-5
154
60-4
151
59-7
149
59-3
148
2-6
64-5
168
63-6
166
62-7
163
61-4
160
60-7
158
60-2
157
2-7
65-9
178
64-9
175
63-8
172
62-4
169
61-6
166
61
165
2-8
67-3
188
66-2
185
64-9
182
63-3
177
62-4
175
61-8
173
2-9
68-6
199
67-4
195
66
191
64-2
186
63-2
183
62-6
182
3
69-8
209
68-5
206
67
201
65-1
195
64
192
63-3
190
31
71
220
69-6
216
68
211
66
205
64-8
201
64
198
3-2
72-2
231
70-7
226
68-9
221
66-8
214
65-5
210
64-7
207
3-3
73-4
242
71-7
237
69-8
230
67-5
223
66-2
219
65-4
216
3-4
74-5
253
72-8
247
70-7
240
68-3
232
66-9
228
66
224
3-5
75-6
265
73-7
258
71-6
251
69
242
67-5
236
66-6
233
3 6
76-6
276
74-6
269
72-4
261
69-7
251
68-1
245
67-2
242
3-7
77-7
287
75-6
281
73-2
271
70-4
260
68-7
254
67-7
251
3-8
78-7
299
76-5
290
74
281
71
270
69-3
263
68-2
259
3-9
79-6
311
77-4
302
74-7
291
71-6
279
69-9
273
68-8
268
4
80-6
322
78-1
313
75-4
302
72-2
289
70-4
282
69-2
277
222
HYDRAULICS
Table XLII. — Manning's Co-efficients.
Values of Kutter's N.
^R
•0U9
■010
■oil
•012
•013
•016
•017
■020
•0225
•025
•0276
•030
•035
■4
121
109
98
91
81
73
64
55
49
44
40
36
31
•5
131
118
106
98
91
79
69
59
52
47
43
39
34
•6
140
125
113
104
97
84
72
63
56
50
45
42
35
■7
147
132
119
110
102
88
80
66
59
53
48
44
38
•8
153
138
124
115
106
92
81
70
61
55
50
46
39
■9
159
143
129
120
111
96
84
72
64
57
52
48
41
10
165
149
134
124
114
99
87
74
66
59
54
50
43
1-1
170
153
138
128
118
102
90
77
68
61
56
51
44
1-2
176
158
142
132
122
106
93
79
70
63
53
53
45
1-3
180
162
li'6'
135
125
108
95
81
72
65
59
54
46
1-4
185
167
150
139
128
111
98
84
74
67
61
56
48
1-5
190
170
154
142
131
114
101
86
76
68
62
57
49
1-6
194
173
157
145
134
116
103
87
78
70
63
58
50
1-7
198
178
161
149
137
119
105
89
79
71
65
59
51
1-8
201
180
163
151
140
121
108
91
81
73
66
61
52
1-9
205
184
166
154
142
123
109
92
82
74
67
62
53
20
208
187
169
156
144
125
no
94
83
75
68
62
54
2-1
212
190
171
159
147
127
113
95
85
76
69
64
54
2-2
215
193
174
161
149
129
114
97
86
78
70
65
55
2-3
218
196
177
163
151
131
116
98
87
79
71
66
56
2-4
221
199
179
166
153
133
117
100
88
80
72
66
57
2-5
224
202
182
168
155
135
119
101
90
81
73
67
58
2-6
227
204
184
170
157
136
120
102
91
82
74
68
58
2-7
230
207
186
172
159
138
122
104
92
83
75
69
59
2-8
233
209
189
174
161
140
123
105
93
84
76
70
60
2-9
235
212
191
177
163
141
125
106
94
85
77
71
60
30
238
214
193
179
165
1-J3
126
107
95
86
78
71
61
31
241
217
195
180
167
144
128
108
96
87
79
72
62
3-2
243
219
197
183
168
146
129
no
97
88
80
73
63
3-3
246
221
199
184
170
147
130
111
98
88
81
74
63
3-4
248
223
201
186
172
149
132
112
99
89
81
74
64
3-5
251
226
203
188
174
150
133
113
100
90
82
75
65
3-6
253
228
205
190
175
152
134
114
101
91
83
76
65
3-7
255
230
207
192
177
153
135
115
102
92
84
77
66
3-8
258
232
209
193
178
154
137
n6
103
93
84
77
66
39
260
234
211
195
180
156 138
117
104
94
85
78
67
4-0
262
236
212
197
181
167 139
118
105
94
86
79
67
OPEN CHANNELS — UNIFOEM FLOW
223
TABLES OF SECTIONAL DATA.
Rectangular and Trapezoidal Sections.
For a bed-width intermediate to those given it is only necessary, in order to
find A, to multiply D by the difference in width and add or subtract the
result. Thus, for bed 43 ft., slope i to 1, and depth 3-75 ft., ^ = 175-8
-3'75 x2 = 168"3 : ^Ji changes so slowly that the correct figure can be
interpolated by inspection. For the above section it is I'Sl.
Widths outside the range of the tables. — To find JB for a width W and
W D
depth D, look out sJR for width -r- and depth -j and multiply by 2, or for
W D
q- and „ and multiply by 3. Interpolations can also be made on this
principle. For instance, the figures for a bed of 12-5 feet can be found
trom those for a 50-feet bed.
For side-slopes ofi to 3 and 3 to 4. — A and ,JR are the same respectively
as for a rectangular section and a J to 1 section of the same mean width.
Thus for a channel of bed 21 feet, side-slopes 4 to 3, and depth 3 feet, the
mean width is 25 feet, and .4 = 75, J li =1-56. For a bed- width of 11 feet,
side-slopes 3 to 4, and depth 4 feet, the mean width is 14 feet, wh'ch is the
same as for a channel with bed 12 feet, side-slopes J to 1, and depth 4 feet.
j4 = 56 and ,^i? = l'64. These rules can be conveniently applied when the
mean widths are whole numbers. For other cases interpolations can be used.
For streams of very shallow section ( W very great in proportion to D)
JR is nearly independent of the ratio of the side-slopes, and depends
practically on the mean width only.
Table XLIII. — Sectional Data for Open Channels,
Rectangular Sections.
Depth
Bed 1 foot.
Bed 2 feet.
Bed S feet.
Bed 4 feet.
Bed 6 feet.
of
Water.
A
VR
A
V-S
A
V-K
A
VR
A
VR
Feet.
•5
■5
•5
I
•56
1^5
■61
2
■63
2-5
•65
■75
•75
•55
1-5
■66
2^25
•71
3
■74
3 75
•76
1
1
•58
2
•71
3
■77
4
■82
5
•85
1-25
1-25
■6
2-5
•74
3^75
■83
5
■88
6^25
•91
1-5
1-5
•61
3
■78
4^5
■87
6
■93
7-5
•97
1-75
1-75
•62
3^5
•8
5^25
•9
7
•97
8^75
I'Ol
2
2
•63
4
•82
6
■93
8
1
10
1-05
2-25
2-25
•64
4^5
•83
6^75
•95
9
103
11-25
ro9
2 '5
2-5
•65
5
•84
7-5
•97
10
ro5
12^5
M2
2-75
2-75
•65
5^5
•86
8^25
•99
11
1-08
13^75
1-U
3
3
■66
6
•87
9
1
12
11
15 ■
M7
3-25
6^5
•87
9-75
roi
13
111
16^25
M9
3-5
7
•88
10-5
ro2
14
M3
17^5
1-21
3-75
...
7^5
•89
11^25
1-03
15
1^14
18-75
1^23
4
8
•89
12
1^04
16
1-15
20
1-24
4-25
...
12^75
1^05
17
117
21 ^25
1-25
4-5
..•
135
1^07
18
M9
22^5
1-27
4-75
14-25
ro7
le
1-19
23-75
1-28
5
...
...
15
1-07
20
1^2
25
1-29
224
HYDRAULICS
Table XLIII. — Continued. {Rectangular.)
Deptli
Bed teet.
Bed 7 feet.
Bed 8 feet.
Bed 10 feet.
Bed 12 feet.
of
Water.
A
VB
A
a/R
A \/R
A
Vfl
A
•/ii
Feet.
■5
3
■65
3-5
-66
4 -67
5
•67
6
•68
•75
4-5
•78
5^25
-79
6 -8
7-5
•81
9
•82
1
6
■87
7
-88
8 -8
10
•91
12
•93
1-25
7-5
■94
8^75
-96
10 -83
12-5
1
15
1-02
1-5
9
1
10-6
1-03
12 1-04
15
r07
18
1-1
1-75
10-5
1^05
12-25
1-08
14 1-1
17-5
114
21
117
2
12
1^1
14
113
16 1-15
20
1-2
24
1-22
2-25
13-5
113
15-75
1-J7
18 1-2
22-5
1-25
27
1-28
2-5
15
117
17-5
1 21
20 1-24
25
129
30
1-33
2-75
16-5
\-2
19-25
1-24
22 1-28
27-5
133
33
1-37
3
18
123
21
1-27
24 1-31
30
1-37
36
1-41
3-25
19-5
1^25
22-75
1-3
26 1-34
32-5
1-4
39
1-45
3-5
21
1^^28
24-5
1-32
28 1-37
35
1-43
42
1-48
3-75
22-5
1-3
26-25
1-35
30 1-39
37-5
1-46
45
1-52
4
24
131
28
1-37
i2 1 -41
40
1-49
48
1-55
4-25
25-5
133
29-75
1-39
34 1 -44
42-5
1-52
51
1-58
4-5
27
134
31-5
1-4
36 1 -46
45
1-64
54
1-6
4-75
28-5
1-36
33-25
1-42
38 1-47
47-5
156
57
1-63
5
30
1-37
35
1-44
40 1-49
50
158
60
1-65
5-25
31-5
1^39
36-75
1-45
42 1-51
52-5
1-6
63
1-67
5-5
33
139
38-5
1-46
44 1 -52
55
1-62
66
1-69
5-75
34-5
1^4
40-25
1-48
46 1 -54
57-5
1-64
69
1-71
6
36
l-H
42
1-49
18 1 -55
60
1-65
72
1-73
6-25
•*.
62-5
1-67
75
1-75
6-5
.,
..
65
1-68
78
1-77
6-75
...
67-5
1-69
81
1-78
7
...
70
1-71
84
1-8
7-25
.,
• •*
72-5
1-72
87
1-81
7-5
75
1-73
90
1-83
7-75
..
77-5
1-74
93
1-84
8
...
SO
1-75
96
1-85
OPEN CHANNELS — UNIFORM FLOW
225
Table XLIII. — Continued. (Rectangular.)
Bod 14 feet.
Bed 16 feet.
Bed 18 feet.
Bed 20 feet.
Bed 26 feet.
' Depth
of
Water.
A
v'iJ
A
VR
A
^B
A
VX
A
Vfi
Feet.
•5
I
•68
8
•69
9
•73
10
-69
12-5
-68
•75
10-5
•82
12
•83
13-0
•83
15
-84
18-8
-84
1
14
•94
16
•94
18
•95
20
-95
25
-96
1-25
17-5
ro3
20
r04
22-5
1^05
25
1-05
31-3
1-07
1-5
21
M2
24
M2
27
M3
30
1-14
37-5
1-16
1-75
24-5
M8
28
1-2
31-5
1-21
35
1-22
43-8
1-24
2
28
1-25
32
1^27
36
r28
40
1-29
50
1-31
2-25
31-5
IS
36
133
40^5
r34
45
1-36
56-3
1-38
2-5
35
136
40
1^38
45
1-4
50
1-41
62-5
1-44
2-75
38-5
r4
44
1-43
49^5
1-45
55
1-47
68-8
1-5
3
42
1^45
48
1-48
54
1-5
60
1-52
75
1-56
3-25
45-5
r49
52
152
58^5
1-55
65
1-57
81-3
1-61
3-5
49
1-53
56
r56
63
1-59
70
1-61
87-5
1-65
3-75
52-5
1^56
60
1-6
67-5
1-63
75
1-65
93-8
1-7
4
56
re
64
1-63
72
1-66
80
1-69
100
1-75
4-25
59-5
1-63
68
r67
76^5
1-7
85
1-73
106-3
1-78
4-5
63
im
72
1-7
81
1-73
90
1-76
112-5
1-82
4-75
66-5
1^68
76
r73
85^5
1-76
95
1-79
118-8
1-82
5
70
1^71
80
1-16
90
1-79
100
1-83
125
1-89
5-25
73-5
1 73
84
1-78
94^5
1-82
105
1-86
131-3
1-92
5-5
77
1^76
88
\-s
99
1-85
110
1-89
137-5
1-95
5-75
80 '5
1-18
92
1-83
103^5
1-87
115
1-91
143-8
1-99
6
84
1-8
96
1^85
108
1-9
120
1-94
150
2-02
6-25
87-5
1-82
100
1-87
112^5
1-92
125
1-96
156-3
2-04
6-5
91-
1^84
104
1-89
117
1-94
130
1-98
162-5
2-07
6-75
94-5
1-85
108
1^91
121 •S
1-96
135
2-01
168-8
2-09
7
98
1^87
112
1-93
126
1-98
140
2-03
175
2-11
7-25
101-5
1'89
116
1-95
130^5
2
145
2-05
181-3
2-14
7-5
105
1-9
120
1-97
135
2-02
150
2-07
187-5
2-17
7-75
l'08-5
1^92
124
1-98
139-5
2-04
155
2-09
193-3
2-19
8
112
1-93
128
2
144
2-06
160
2-11
200
2-21
8-25
148-5
2-07
165
2-13
206-3
2-23
8-5
153
2 09
170
2-14
212-5
2-25
8-75
157-5
2-11
175
216
218-8
2-27
9
162
2-12
180
2-18
225
2-29
9-25
166-5
2-14
185
219
231-3
2-31
9-5
171
2-15
190
2-21
237-5
2-32
9-75
175-5
2-16
195
2-22
243-8
2-34
10
180
2-18
200
2-24
250
2-36
226
HYDRAULICS
Table XLIII. — Continued. {Rectangiilar)
Depth
of
Water.
Bod 30 feet.
Bed 35 feet.
Bed 40 feet.
Bed SO feet.
Bed 60 feet.
A
VB
A
s/R
A
-^R
A
VR
A
VJJ
Feet.
1
30
-97
srj
•97
40
•98
50
-98
60
-98
1-5
35
1-17
52-5
1-18
60
1-18
75
1-19
90
1-2
2
60
1-33
70
1-34
80
1-35
100
1-36
120
1-37
2-25
67-5
1-39
78-8
1-41
90
1-42
112-5
1-44
135
1-45
2-5
75
1-46
87-5
1-48
100
1-49
125
1-51
150
1:52
2-75
82-5
1-53
96-3
1-54
110
1-56
137 -5
1-57
165
1-59
3
90
1-58
105
1-6
120
1-62
150
1-64
180
l-6o
3-25
97-5
1-63
113-8
1-66
130
1-67
162-5
, 1-7
195
1-71
3-5
105
1-68
122-5
1-71
140
1-73
175
1-75
210
1-77
3-75
112-5
1-73
131-3
1-75
150
1-76
187-5
1-81
2-25
1-83
4
120
1-78
140
1-78
160
1-83
200
1-86
240
1-88
4-25
127-5
1-82
148-8
1-85
170
1-87
212-5
1-91
•255
1-93
4 5
135
1-86
157-5
1-89
180
1-92
225
1-95
270
1-98
4-75
142-5
1-9
166-3
1-93
190
1-96
237-5
2
285
2-03
5
150
1-94
175
1-99
200
2
250
2-04
300
2-07
5-25
157-5
1-97
183-8
2-01
210
2-04
262-5
2-08
315
2-11
5-5
165
2-01
192-5
2-04
220
2-08
275
2-12
330
216
5-75
172-5
2-04
201-3
2-08
230
211
287-5
2-16
345
2-2
6
180
2-07
210
2-11
240
2-15
300
2-2
360
2-24
6-25
187-5
2-1
218-8
2-15
250
2-18
312-5
2-24
375
2-27
6-5
195
2-13
227-5
2-18
260
2-22
325
2-27
390
2-31
6-75
202-5
2-16
236-3
2-21
270
2-25
337-5
2-31
405
2-35
7
210
2-18
245
2-24
280
2-28
350
2-34
4-20
2-38
7-25
217-5
2-21
253-8
2-26
290
2-31
362-5
2-37
435
2-42
7-5
2-25
2-24
262-5
2-29
300
2-34
375
2-4
450
2-45
7-75
232-5
2-26
271-3
2-32
310
2-37
387-5
2-43
465
248
8
240
2-28
280
2-34
320'
2-39
400
2-46
480
251
8-25
247-5
2-31
288-8
2-37
330
2-42
412-5
2-49
495
2-54
8-5
255
2-33
297-5
2-39
340
2-44
4-2r)
2-52
510
2-57
8-75
262-5
2-35
306-3
2-42
350
2-47
437-6
2-.V>
525
2-6
9
270
2-37
315
244
360
2-49
450
2-57
540
2-63
9-25
277-5
2-39
323-8
2-46
370
2 -02
462-5
2-6 .
555
2-66
9-5
285
2-41
332-5
2-48
380
2 -."14
475
2-02
570
2-69
9-75
292-5
2-43
341-3
2-5
390
2-.i6
487-5
2-65
5S5
2-71
10
300
2-4.5
350
2-.V2
400
2-58
500
2-67
600
2-74
10-5
625
2-72
630
2-79
11
550
2-76
660
2-84
11-5
• ••
>*■
575
2-81
690
2-88
12
-•*
600
2-85
720
2-93
OPEN CHANNELS — UNIFORM FLOW
227
Table XLIII. — Continued. (Bedangular.)
Depth
of
Water.
Bed 70 feet.
Bed 80 feet.
Bed 90 feet.
Bed 100 feet.
Bed 120 feet.
A
V-R
A
^/i^
A
VB
A
Vli
A
VB
Feet.
1
70
80
90
100
120
1-5
105
120
135
150
180
2
140
160
180
200
240
2-25
157-5
180
202-5
225
270
2-5
175
200
225
250
300
2-75
192-5
220
2*7-5
275
330
3
210
240
270
300
360
3-25
227-5
260
292-5
325
390
3-5
245
280
315
350
420
3-75
262-5
300
337-5
375
450
4
280
320
360
400
480
4-25
297-5
340
382-5
425
510
4-5
315
360
405
450
540
4-75
332-5
380
427-5
475
570
5
350
400
450
500
600
5-25
367-5
420
472-5
525
630
5-5
385
440
495
550
660
5-75
402-5
460
517-5
575
690
6
420
480
540
600
720
6-25
437-5
500
562-5
625
750
6-5
455
520
585
650
780
6-75
472-5
540
607-5
675
810
7
490
560
630
700
840
7-25
507-5
580
652-5
725
870
7-5
525
600
675
750
900
7-75
542-5
620
697-5
775
930
8
560
640
720
800
960
8-25
577-5
660
742-5
825
990
8-5
595
680
765
850
1020
8-75
612-5
700
787-5
875
1050
9
630
720
810
900
1080
9-25
647-5
740
8;!2-5
925
1110
9-5
665
760
855
950
1140
9-75
682-5
780
877-5
975
1170
10
700
800
900
1000
1200
10-5
735
840
945
1050
1260
11
770
880
990
1100
1320
11-5
805
920
1035
1150
1380
12
840
960
1080
1200
1440
228
HYDRAULICS
Table XLIV. — Sectional Data for Open Channels.
Trapezoidal Sections — Side-slopes \ to \.
Dcpl.h
of
Water.
Bed 1 foot.
Bed 2 feet.
Bed 8 feet.
Bed 4 feet.
Bed 5 feet.
A
Vli
A
Vlt
A
VR
A
Vll
A
VK
Feet.
•5
•63
■54
1-13
•60
1-63
•63
2-13
■64
263
■65
•75
103
•62
1-78
•69
2-53
•73
3-28
■76
4^03
■77
1
1-5
•68
2'5
•77
3-5
•82
4-5
•85
5^5
■87
1-25
2-03
•73
^•28
•83
4-53
•88
5-78
■92
703
■95
1-5
2-63
■78
4^13
•88
5-63
•94
7^13
■98
8 63
102
1-75
3-28
•82
5 •OS
-92
6-78
■99
8^53
1^04
10-28
108
2
4
•86
6
•96
8
103
10
109
12
113
2-25
4-78
•89
7^03
1
9-28
1-07
11-53
lis
1378
1-17
2-5
5-63
•92
8 •IS
1-03
10-63
Ml
1313
117
15 63
121
2-75
6-53
■95
9-28
1-07
12 03
1^15
14^78
1-21
17 53
125
3
7-5
•99
10-5
1-1
13-5
M8
16-5
124 ' 195
129
3-25
...
\\-1S.
M3
15-03
1-21
18 28
1-27
21-53
ISS
3-5
...
13-13
1-16
16^63
1-24
2013
rso
23-63
136
3-75
...
14-53
1-18
18-28
1-27
22^03
133
25-78
139
i
16
1-21
20
1'29
24
1^36
28
142
1
4-25
...
...
21 •7S
1-32
26 ■OS
1S9
30^28
1
4-5
...
23-63
1 -3.-)
2813
1-41
32-63
l-i-
4'75
...
...
'l-rWi
1-37
30-28
144
35-03
15
5
'27-5
1-39
32-5
146
37-5
r52
OPEN CHANNELS — UNIFORM FLOW
229
Table XLIV. — Continued. (| to 1.)
Depth
Bed 6 feet.
Bed 7
feet.
Bed 8 feet.
Bed 9 feet.
Bed 10 feet.
of
Water.
A
^R
A
V-fi
A
V-R
A
VR
A
s/R
Feet
■5
3-13
-66
3-63
•67
4-13
•67
4-63
-68
5-13
•68
•75
4-78
•78
5-53
-79
6-28
-8
7-03
-81
7-78
-81
1
6-5
-89
7-5
-9
8-5
•91
9-5
-92
10-5
-93
1-25
8-28
■97
9-53
-99
10-78
1
12-03
1-01
13-28
1-02
1-5
1013
1-04
11-63
1-06
13-13
1-07
14-63
1-09
16-13
1-1
1-75
12 03
1-1
13-78
1-12
15-53
1-14
17-28
116
19-03
1-17
2
14
1-16
16
1-18
18
1-2
20
1^22
22
1-23
2-25
16-03
1-21
18-28
ri'3
20-53
1-25
22-78
1-28
25 03
1-29
2-5
1813
1-25
20-63
1-28
23-13
1-3
25-63
1-33
28-13
1-34
2-75
20-28
1-29
23-03
1-32
25-78
1-35
28-53
1-38
31-28
1-39
3
22-5
1-33
25-5
1-36
28-5
1-39
31-5
1-42
34 5
1-44
3-25
24-78
1-37
28-03
1-4
31-28
1-43
34-53
1-46
37-78
1-48
3-5
27-13
1-4
30-63
1-44
34-13
1-47
37-63
1-5
41-13
1-52
3-75
29-23
1-43
33-28
1-47
37-03
1-5
40-78
1-54
44-53
156
4
32
1-46
36
1-5
40
1-54
44
1-57
48
1-59
4-25
34-53
1-49
38-78
1-53
43-03
1-57
47-28
1-6
51-53
1^63
4-5
37-13
1-52
41-63
1-56
46-13
1-6
50-63
1-63
55-13
166
4-75
39-78
1-55
44-53
1-59
49-28
1-63
54-03
1-66
58-78
169
5
42-5
1-57
47-5
1-62
52-5
1-65
57-5
1-69
62-5
1-72
5-25
45-28
1-6
50-53
1-65
55-78
1-68
6103
1-72
65-28
1-75
5-5
48-13
1-62
53-63
1-67
59-13
1-71
64-63
1-74
70-13
1-77
5-75
51-03
1-65
56-78
1-09
62-53
1-73
68-28
1-77
74-03
1-8
6
54
1-67
60
1-71
66
1-76
72
1-79
78
1-82
6-25
...
...
...
82-03
1-85
6-5
...
86-12
1-87
6-75
• •.
...
90-28
1-9
7
94-5
1-92
7-25
98^78
1-94
7-5
1031
1-96
7-75
...
107^53
1-98
8
...
...
112
2
230
HYDRAULICS
Table XLIV—
-Conii/n
ued.
(|toi
1-)
Bed 12 feet.
Bed 14 feet.
Bed 16 feet.
Bed 18 feet.
Bed 20 feet.
Depth
of
Water.
A
V'B
A
VR
A
VR
A
^R
A
VR
Feet.
•5
6-1
•68
7-1
-69
8-1
•69
9-1
-69
10-13
-69
■75
9-3
•82
10-8
■83
12-3
-83
13-8
•84
15-28
-84
1
12-5
-94
14-5
-94
16-5
•95
18-5
•96
20-5
-96
1-25
15-8
1-03
18-3
1-05
20-8
1-05
23-3
106
25-8
1-06
1-5
19-1
1-12
221
1-13
25-1
1^14
28-1
115
31 1
115
1-75
22-5
1-19
26
1-2
29-5
1^22
33-3
123
36 5
1-23
2
26
1^26
30
1-27
34
V29
38
13
42
1-31
2-25
29-5
1-32
34
1-33
38-5
135
43
r37
47^5
1-38
2-5
331
1-37
38-1
1-39
43-1
1-41
48-1
143
531
1-44
2-75
36-8
1-42
42-3
1-45
47-8
1-47
53 3
1-49
58-8
1-5
3
40-5
1-47
46-5
1-5
52-5
1-52
58-5
1-54
64-5
1-55
3-25
44-3
1-52
50-8
1-55
57-3
1-57
63-8
1-59
70-3
1-6
3-5
48-1
1-56
55-1
1-59
621
1-61
691
1-64
76-1
1-65
3-75
52
1-6
59-5
1-63
67
1-66
74-5
1-68
82
17
4
56
1-64
64
1-67
72
!•?
80
1-72
88
174
4-25
60
1-67
68-5
1-71
76
1^74
84 5
176
94
1-79
4-5
64-1
1-7
731
1-74
82-1
1-78
911
1-8
100-1
183
4-75
68-3
1-74
77-8
1-78
87-3
1-81
9K-8
r84
106-3
1^86
5
72-5
1-77
82-5
1-81
92-5
1-84
102-5
1-87
112-5
1-9
5-25
76-8
1-8
87-3
1-84
97-8
1-88
108 3
1-91
118-8
1-94
5-5
81-1
1-83
92-1
1-87
103-1
1-91
1141
r94
125-1
1-97
5-75
85-5
1-86
97
1-9
108-5
1-!14
120
197
131-5
2
6
90
1-88
102
1-93
114
1-97
126
2
138
2 03
6-25
94-5
1-91
107
1-96
119-5
2
132
2 03
1445
2-06
6-5
991
1-93
112-1
1-98
125-1
2-02
138 1
206
151-1
2 09
6-75
103 8
1-96
117-3
2-01
130-8
2 05
144-3
2 09
157 8
212
7
7-25
108-5
1-98
122-5
2 03
1.36-5
2-08
150^5
211
164-5
215
113-3
201
127-8
206
142-3
211
156-8
214
171-3
2-18
7 -5
1181
2-03
133 1
2 08
148-1
213
163-1
•217
178-1
2'*2
7-75
123
2-05
138-5
2-1
154
215
169-5
219
185
2 23
g
128
2-07
144
2-12
160
2-17
176
2 21
192
2-25
8-25
8-5
8-75
9
9-25
9-5
9-75
10
182-5
2-24
199
2 -28
191 2
2-26
208-2
2-3
195-8
2-28
213-3
2-32
'202-5
2-3
2-20-5
2-34
209-3
2-33
2-27-8
2-37
...
2161
2-35
235 1
2-39
223
2-37
242-5
2^41
.. .
:::
230
2-39
250
243
OPEN CHANNELS — UNIFORM FLOW
231
Table XLIY .—Coniinued. (J to 1.)
Depth
of
Water.
Bed 26 feet.
Bed SO feet.
Bed 35 feet.
Bed 40 feet.
Bed 45 feet.
A
V-B
A
V-R
A
VR
A
VS.
A
VR
Feet.
1
25-5
-97
30-5
-97
35-5
•98
40-5
•98
45-5
■98
1-5
38-6
1-17
46-1
1-18
53-6
1-18
61
1-19
68-6
1-19
2
52
1-33
62
1-34
72
1-35
82
1-36
92
1-36
2-25
58-8
1-4
70
1-41
81-3
1-42
92-5
1-43
103-8
1-44
2-5
65-6
1-46
78-1
1-48
90-6
1-49
103-2
1-5
115-6
1-51
2-75
72-5
1-52
86-3
1-54
100
1-56
113-8
1-57
127-5
1-58
3
79 5
1-58
94-5
1-6
109-5
1-62
124-5
1-63
139-5
1-64
3-25
86-5
1-64
102-8
1-66
119
1-63
135-3
1-69
151-5
1-7
3-5
93-6
1-69
111-1
1-71
123-6
1-73
146-1
1-75
163-6
1-76
3-75
100-8
1-74
119-5
1-76
138 -3
1-79
157
1-8
175-8
1-82
4
108
1-78
128
1-81
148
1-84
168
1-85
188
1-87
4-25
115-3
1-83
136-5
1-86
157-8
1-89
179
1-9
200-3
1-92
4-5
122-6
1-87
145-1
1-9
167-6
1-93
190-1
1-95
212-6
1-96
4-75
130
1-91
153-8
1-95
177-5
1-97
201-3
2
225
2-01
5
137-5
1-95
162-5
1-99
187-5
2-01
212-5
2-04
237-5
2-06
5-25
145
1-99
171-3
2-03
197-5
2 05
223-8
2-08
250
21
5-3
152-6
2 02
180-1
2-06
207-6
2-1
235 1
2-12
262-6
2-14
5-75
160-3
2-06
189
21
217-8
214
246-5
2-16
275-3
2-18
6
168
2-09
198
2-14
•228
2-17
258
2-2
288
2-22
6-25
175-8
2-12
207
2-17
238-3
2-21
269-5
2-24
300-8
2-26
6-5
183-6
2-15
216-1
2-2
248-6
2-24
281-1
2-27
313-6
2-3
6-75
191-6
2-19
225-3
2-24
259-1
2-28
292-8
2-31
326-6
2-34
7
199-5
2-22
234-5
2-27
269-5
2-31
304-5
2-34
339-5
2-37
7-25
207-5
2-25
243-8
2-3
280
2-34
316-3
2-37
352-5
2-4
7-5
215-6
2-27
253-1
2-33
290-6
2-37
328-1
2-4
365-6
2-43
7-75
223-8
2-3
262-5
2-36
301-3
2-4
340
2-44
378-8
2-47
8
232
2-33
272
2-38
312
2-43
352
2-47
392
2-5
8-25
240-3
2-36
281-5
2-41
322-8
2-46
364
2-5
405-3
2-53
8-5
248-6
2-38
291-1
2-44
333-6
2-49
376-1
2-52
418-6
2-56
8-75
257
2-4
300-8
2-47
344-6
2-52
388-3
2-55
432-1
2-59
9
265-5
2-43
310-5
2-49
355-5
2-i54
400-5
2-58
445-5
2-62
9-25
274-1
2-45
320-3
2-52
366-6
2-57
412-8
2-61
459-1
2-64
9-5
282-6
2-47
330-1
2-54
377-6
2-59
425-1
2-63
472-6
2-67
9-75
291-3
2-5
340
2-57
388-8
2-62
437-5
2-66
486-3
2-7
10
300
2-52
350
2-59
400
2-64
450
2-69
500
2-72
10-5
475-1
2-74
527-6
2-78
11
500-5
2-78
555-5
2-83
I1'5
526-1
2-83
583-6
2-87
12
...
...
552
2-87
612
2-92
232
HYDRAULICS
Table XLIV. — Contvmtd. (J to 1.)
Bed 60 feet.
Bed 00 feet.
Bed ro feet.
Bed 80 feet.
Bed 90 feet.
Depth
of
Water.
A
v-Zi
A
^R
A
Vii
A
VR
A
VJJ
Feet.
1
51-5
■98
60-5
-99
70-5
-99
80-5
•99
90-5
-99
1-5
76-1
1-19
91-1
1-2
106-1
1-21
121 1
1-21
136-1
1-21
2
102
1-37
122
1-38
142
1-38
162
1-38
182
1-39
2-25
115
1-45
137-5
1-46
160
1-46
182-5
1-47
205
1-47
2-5
128-1
1-52
153-1
1-53
1781
1-54
203-1
1-54
228-1
1-54
2-75
141-3
1-59
168-8
1-6
196-3
1-61
224-8
1-61
252-3
1-62
3
154-5
1-65
184-5
1-66
214-5
1-67
244-5
1-68
274-5
1-68
3-25
167-8
1-71
200-3
1-73
232-8
1-74
265-3
1-74
297-8
1-75
3-5
181-1
1-77
216-1
1-78
251-1
1-8
286-1
1-8
321-1
1-81
3-75
194-5
1-83
232
1-84
269-5
1-86
307
1-86
344-5
1-87
4
208
1-88
248
1-9
288
1-91
328
1-92
368 .
1-93
4-25
221-5
1-93
264
1-96
306-5
1-96
349
1-98
391-5
1-99
4-5
235-1
1-98
280-1
2
325-1
2-02
370-1
2-03
4161
2-04
4-75
248-8
2-03
296-3
2-05
343-8
2-07
391-3
2-08
438-8
2-09
5
262-5
2-07
312-5
2-1
362-5
2-11
412-5
213
462-5
2-14
5-25
276-3
2-12
328-8
2-15
381-3
2-16
433-8
2-18
486-3
2-19
5-5
290-1
2-16
345-1
2-18
400-1
2-2
455-1
2-22
5101
2-23
5-75
304
2-2
361-5
2-23
419
2-25
476-5
2-26
534
2-28
€
318
2-24
378
2-27
438
2-29
498
2-31
558
2-32
6-25
332
2-28
394-6
2-31
457
2-33
519-5
2-35
582
2-37
6-5
346-1
2.32
411-1
2-35
476-1
2-37
541-1
2-39
606-1
2-41
6-75
360-3
2-36
427-8
2-39
495-3
2-41
562-8
2-43
630-3
2-45
7
374-5
2-39
444-5
2-42
514-5
2-45
584-5
2-47 654-5
2-49
7-25
388-8
2-43
461-3
2-46
533-8
2-49
606-3
2-51
678-8
2-53
7-5
403-1
2-46
478-1
2-5
653-1
2-52
628-1
2-55
703-1
2-57
7-76
417-5
2-49
495
2-53
572 -i5
2-56
650
2-59
7-27-5
2-6
8
432
2-52
512
2-56
592
2-6
672
2-62
752
2-64
8-25
446-5
2-55
529
2-59
6U-5
2-63
694
2-66
776-5
2-68
8-5
461-1
2-58
546-1
2-63
631-1
2-66
716-1
2-69
701-1
2-71
8-75
475-8
2-61
563-3
2-66
650-8
2-7
738-3
2-73
825-8
2-75
9
490-5
2-64
580-5
2-69
670-5
2-73
760-5
2-76
850-5
2-78
9-25
505-3
2-67
597-8
2-72
690-3
2-76
782-8
2-79
875-3
2-81
9-5
520-1
2-7
015-1
2-75
710-1
2-79
805-1
2-82
900-1
2-84
9-75
535
2-73
632-6
2-78
730
2-82
8-27 -5
2-85
925
2-88
10
550
2-76
erio
2-81
750
2-85
850
2-88
950
2-91
10-5
fiSO-I
2-81
686-1
2-80
790-1
2-91
895-1
2-94
1000
2-97
11
«10-5
2-86
720-5
2-92
830-5
2-96
940-6
3
1050
3-03
11-5
641-1
2-91
766-1
2-07
871-1
3-02
986-1
3-05
1101
3-08
12
072
2-i«;
792
3 02
912
307
1032
3-11
1152
3-14
OPEN CHANNELS — UNIFORM FLOW
233
Table XLIY.— Continued. (J to 1.)
Depth
ot
Water.
Bed 100 feet.
Bed 120 feet.
Bed 140 feet.
Bed 160 feet.
A
VR
A
Vlt
A
V-K
A
VR
Feet.
1
100-5
■99
120-5
-99
140-5
•99
160-5
-99
1-5
1511
1-21
181-1
1-21
211-1
241-1
1-4
2
202
1-39
242
1-39
282
1-4
.322
1-56
2-25
227-5
1-47
272-5
1-47
2-5
253-1
1-55
303-1
1-55
3531
1-56
403-1
2-75
278-8
1-62
333-8
1-62
388-8
1-63
443-8
1-63
3
304-5
1-69
364-5
1-69
424-5
1-7
484-5
1-7
3-25
330-3
1-76
395-3
1-76
460-3
1-77
525-3
1-77
3-5
356-1
1-82
426-1
1-82
496-1
1-83
566-1
1-83
3-75
382
1-88
457
1-88
532
1-89
607
1-9
4
408
1-94
488
1-94
568
1-95
648
1-96
4-25^
434
1-99
519
2
604
2-01
689
2-02
4-5
460-1
2-04
550-1
2-05
640-1
2-06
730-1
2-07
4-75
486-3
2-1
581-3
211
676-3
2-12
771-3
213
5
512-5
2-15
612-5
216
712-5
2-17
812-5
2-18
5-25
538-8
2-2
643-8
2-21
748-8
2-22
853-8
2-23
5-5
565-1
2-24
675-1
2-25
785-1
2-26
895-1
2-28
5-75
591-5
2-29
706-5
2-3
821-5
2-31
9.36-5
2-32
6
618
2-33
738
2-35
858
2.36
978
2-37
6-25
644-5
2-38
769-5
2-4
894-5
2-41
1020
2-42
6-5
671-1
2-42
801-1
2-44
931-1
2-45
1061
2-46
6-75
697-8
2-46
832-8
2-48
967-8
2-5
1103
2-51
7
724-5
2-5
864-5
2-52
1005
2-54
1145
2-55
7-25
751-3
2-54
896-3
2-56
1041
2-58
1186
2-59
7-5
778-1
2-58
928-1
2-6
1078
2-62
1228
2-63
7-75
805
2-62
960
2-64
1115
2-66
1270
2-68
8
832
2-66
992
2-68
1152
2-7
1312
2-72
8-25
859
2-69
1024
2-72
1189
2-74
13i54
2-76
8-5
886-1
2-73
1056
2-75
1226
2-78
1396
'2-79
8-75
913-3
2-76
1088
2-79
1263
2-82
1438
2-83
9
940-5
2-8
1121
2-83
1301
2-85
1481
2-86
9-25
967-8
2-83
1153
2-86
1338
2-89
1523
2-9
9-5
995 1
2-86
1185
2-89
1375
2-92
1565
2-93
9-75
1023
2-9
1118
2-93
1413
2-96
1608
2-97
10
1050
2-93
1250
2-96
1450
2-99
1650
3
10-5
1105
2-99
1315
3 03
1525
3-05
1735
3-07
11
1161
3-05
1381
3 09
1601
3-12
1821
3-13
11-5
1216
311
1446
3-15
1676
3-18
1906
3-2
12
1272
3-17
1512
3-21
1752
3-24
1992
3-27
234
HYDRAULICS
Table XLV. — Sectional Data for Open Channels
Trapezoidal Sections — Side-slopes I to 1.
Depth
of
Water.
Bed 1 foot.
Bed 2 feet.
Bed 3 feet.
Bed 4 feet.
Bed 5 feet.
■A
VR
A
VR
A
v/JJ
A
VR
A
v'ii
Feet.
■5
•75
•577
1-25
■605
1-75
■629
2-25
-645
2 75
-655
■75
1-31
■652
2^06
■707
2-81
■741
3-56
■763
431
•779
1
2
■723
3
■788
4
■828
5
■856
6
-875
1-25
2-81
•787
4^06
■856
5-31
■901
6 56
■933
"■81
-956
1-5
3-75
■846
5^25
■917
6-75
■965
8^25
1
9-75
1-03
1-75
4-81
■899
6 56
•971
8-31
1-02
1006
1^06
11 81
1-09
2
6
■95
8
r02
10
1-08
12
1-12
14
1-15
2-25
7-31
■996
9 56
1-07
11-81
1-12
1406
M7
1631
1-2
2-5
8-75
1-04
1P25
Ml
13-75
117
16 25
1-21
18-7o
1-25
2-75
10-32
P08
13^06
116
15-81
1-21
18^56
1-26
2131
1-29
3
12
113
15
r2
18
1-25
21
13
24
1-33
3-25
17^06
124
20-31
1-29
23 56
1-34
26 81
1-37
3-5
1925
1-27
22-75
1-33
-26-2.-)
138
29-75
1-41
3-75
21-56
131
25-31
1-36
29-06
1-41
32-Sl
1-45
4
24
1-34
28
1-4
32
145
36
1-49
4-25
...
30-81
1-43
:«-06
r4S
39-31
1-52
4-5
...
33 7r>
1-47
3S"i)
1^51
42-75
1-55
4-75
...
36^81
1-r.
41-56
1 ^54
46-32
1-59
r
40
1 •.-.:!
■).■>
1 ■.vs
50
1-62
OPEN CHANNELS — UNIFORM FLOW
235
Table XLV. — Continued. (1 to 1.)
Bed 6 feet.
Bed 7 feet.
Bed 8 feet.
Bed 9 feet.
Bed 10 feet.
Depth
of
Water.
A
v'B
A
vn
A
VR
A
VR
A
VR
Feet.
•5
3-25
-662
3-75
-667
4-23
-672
4-63
-667
3-25
■678
•75
5-06
-781
5-81
-798
6-56
•805
7-03
-795
8-06
•815
1
7
-891
8
-902
9
-911
10
-919
11
•926
1-25
9 06
-975
10-31
-989
11-56
1
12-81
1-01
14-06
1-02
1-5
11-25
1-05
12-75
1-07
14-25
1-08
15-75
1-09
17-25
1-1
1-75
13-56
1-11
15-31
1-13
17-06
1-15
18-81
1-16
20-56
1-17
2
16
1-17
18
1-19
20
1-21
22
1-23
24
1-24
2-25
18-56
1-23
20-81
1-25
23-06
1-27
25-31
1-28
27-56
1-29
2 -.5
21-23
1-28
23-75
1-3
26-25
1-32
28-75
1-33
31-25
1-35
2-75
24-06
1-32
26-81
1-35
29-56
1-37
32-31
1-39
35-06
1-40
3
27
1-37
30
1-39
33
1-42
36
1-44
39
1-45
3-25
30-06
1-41
33-31
1-43
35-56
1-44
39-81
1-48
43-06
1-5
3-5
33-25
1-45
36-75
1-47
40-25
1-51
43-75
1-52'
47-25
1-54
3-75
36-56
1-48
40-31
1-51
44 06
1-.54
47-81
1-56
51-56
1-58
4
40
1-52
44
1-55
48
1-58
52
1-6
56
1-62
4-25
43-56
1-56
47-81
1-59
52-06
1-61
56-31
1-64
60-56
1-66
4-5
47-25
1-59
51-75
1-62
56-25
1-65
60-75
1-67
65-25
1-69
4-75
51-06
1-62
55-81
1-65
60-56
1-68
65-31
1-71
70-06
1-73
5-5
55
1-65
60
1-68
65
1-71
70
1-74
75
1-76
5-25
59-06
1-68
64-31
1-72
69-56
1-75
74-81
1-77
80-06
1-8
5
63-25
1-71
68-75
1-75
74-25
1-78
79-75
1-8
85-25
1-83
5-75
67-57
1-74
73-32
1-78
79-07
1-81
84-82
1-83
90-57
1-86
6
77
1-77
78
1-8
84
1-83
90
1-86
96
1-89
6-25
...
95-31
1-89
101 -56
1-92
6-5
...
100-7
1-92
107-2
1-94
6-75
106-3
•1-94
113-05
1-97
7
...
112
1-97
119
2
7-25
*••
117-8
2
125-05
2-03
7-5
...
123-8
2-02
131-3
2*06
7-75
...
129-8
2-05
137-55
2-08
8
...
...
136
2-07
144
2-11
236
HYDRAULICS
Table XLV. — Confirmed. (1 to 1.)
Depth
of
Water.
Bed 12 feet.
Bed 14 feet.
Bed 16 feet.
Bed 18 feet.
Bed 20 feet.
A
Vfi
A
VJ2
A
ViJ
A
VR
A
VJt
Feet.
•5
6-25
•682
7-37
•683
8-37
•686
9-25
•477
10-25
■69S
•75
9-56
■823
11-34
-824
12-84
•828
1406
-694
15-56
•839
1
13
■936
15
-944
17
•95
19
-955
21
•959
1-25
16-56
1-03
1906
1-04
21-56
105
24 06
1-06
26-56
106
1-5
20-25
1-12
23-25
113
26-25
1-14
29-25
1-15
32-25
1-15
1-75
24-06
1-19
27-56
1-21
31-06
1-22
34-56
1-23
38-06
1^24
2
28
1-26
32
1-28
36
1-29
40
1-3
44
1-31
2 -23
32-06
1-32
36-56
1-34
4106
1.35
45-56
1-37
50-06
1-38
2-5
36-L'5
1-38
41-25
1-4
46-25
1-42
51-25
1-43
56-25
1-44
2-75
40-56
1-43
46-06
1-45
51-56
1-47
57-06
1-49
6256
1-5
3
45
1-48
51
1-51
57
1-53
63
154
69
1-56
3-25
49-56
1-53
56-06
1-56
62-56
1-58
69 06
1-59
75-56
1-61
3-5
54-25
1-57
61-25
1-6
68-25
1-62
75-25
164
82-25
1-66
3-75
59-06
1-62
66-56
1-65.
74-06
1^67
81-56
1-69
89-06
1-71
4
64
1-66
72
1-69
80
1-71
88
1-73
96
1-73
4-25
69-06
1-7
77-56
1-73
86-06
r75
94-56
177
1031
179
4-5
74-25
1-73
83-25
1-77
92-25
179
101-3
1-81
110-3
1-84
4-75
79-56
1-77
89 06
1-8
98-56
1-83
108-1
1-85
117-6
1-88
5
85
1-8
95
1-84
105
1^87
115
1-89
125
1-91
5-25
90-56
1-84
101-1
1-87
111-6
1-9
122-1
1-93
132-6
1-95
5-5
96-25
1-87
107-3
1-91
118-3
1-94
129-3
1-96
140-3
1-99
5-75
102-1
1-9
113-6
1-94
125-1
1-97
136-6
2
1481
2-02
6
108
1-93
120
1-97
132
2
144
2-03
156
2 -(15
6-25
114-1
1-96
126-6
2
1391
2-03
151-6
206
1641
209
6-5
120-2
1^99
133-3
2 03
146-3
2-06
159-3
2-09
172-3
2^12
6-75
126-6
2^02
140-1
2 06
153-6
2-09
1671
212
180-6
215
7
133
2^05
147
2-09
161
212
175
215
189
218
7-25
139-6
2-07
1541
2-11
168-6
2-15
183-1
2-18
197-6
2 21
7-5
146-3
2-1
161-3
2-14
176-3
218
191-3
2-21
206-3
2-24
7-75
163-1
213
•168-6
2-17
184-1
2-21
199-6
2-24
215-1
2-27
8
160
2-15
176
219
192
2-23
208
2 ■26
224
2-29
8-25
8-^
200-1
2-26
216-6
2-29
233-1
232
...
208-3
2-28
225-3
2-:«
■242-3
235
8-75
216-6
2-31
2341
2-34
■2.-.1-6
2-37
9
225
2-33
243
2-37
261
2-4
9-25
...
233-6
2-3.">
•2;-.2-l
2-39
270-6
2-42
9 5
242-3
2-38
261 -3
2-41
280-3
2-45
9-75
2,-) 1-1
2-4
■270-6
2-44
2901
2-47
10
2G0
2-42
280
2-46
300
2-49
OPEN CHANNELS — UNIFORM FLOW
237
Table XLV. — Continued. (1 to 1.)
Depth
of
Water.
Bed 25 feet.
Bed 30 feet.
Bed 36 feet.
Bed 40 feet.
Bed 46 feet.
.-1
^R
A
^R
A
v«
A
Vii
A
VB
Feet.
1
26
-966
31
•976
36
-976
41
-978
46
-981
1-5
39-75
1-17
47-25
1-18
54-75
1-18
62-25
1-19
69-75
1-19
2
54
1-33
64
1-34
74
1-35
84
1-36
94
1-36
2-25
61-31
1-4
72-56
1-41
83-81
1-42
95-06
1-43
106-3
1-44
2-5
68-75
1-46
81-25
1-47
93-75
1-49
106-3
1-5
118-8
1-51
2-75
76-31
1-53
90-06
1-54
103-8
1-56
117-6
1-57
131-3
1-58
3
84
1-58
99
1-6
114
1-62
129
1-63
144
1-64
3-25
91-81
1-64
108-1
1-66
124-3
1-68
140-6
1-69
15e'8
1-7
3-5
99-75
1-69
117-3
1-77
134-8
1-73
152-3
1-75
169-8
1-76
3'75
107-8
1-74
126-6
1-77
145-3
1-79
164-1
1-8
182-8
1-81
4
116
1-79
136
1-81
156
1-84
176
1-85
196
1-87
4-25
124-3
1-83
145-6
1-86
166-8
1-88
188-1
1-9
209-3
1-92
4-5
132-8
1-88
155-3
1-91
177-8
1-93
200-3
1-95
222-8
1-96
4-75
141-3
1-92
165-1
1-95
188-8
1-97
212-6
1-99
236-3
2-01
5
150
1-96
175
1-99
200
2-02
225
2-04
250
206
5-25
158-8
1-97
185-1
2-03
211-3
2-06
237-6
2-08
263-8
2-1
5-5
167-8
2-03
195-3
2-07
222-8
2-1
250-3
2-12
277-8
2-14
5-75
176-8
2-07
205-6
2-11
234-3
2-14
263-1
2-16
291-8
2-18
6
186
2-11
216
2-15
246
2-18
276
2-2
306
2-22
6-25
195-3
2-14
226-6
2-18
257-8
2-21
289-1
2-24
320-3
2-26
6-5
204-8
2-17
237-3
2-21
269-8
2-25
302-3
2-28
334-8
2-3
6-75
214-3
2-2
248-1
2-25
281-8
2-28
315-6
2-31
349-3
2-34
7
224
2-24
259
2-28
294
2-32
329
2-34
364
2-37
7-25
233-8
2-27
270-1
2-31
306-3
2-35
342-6
2-37
378-8
2-41
7-5
243-8
2-3
281-3
2-34
318-8
2-38
356-3
2-41
393-8
2-44
7-75
253-8
2-33
292-6
2-37
.331-3
2-41
370-1
2-45
408-8
2-47
8
264
2-35
304
2-4
344
2-44
384
2-48
4-24
2-5
8-25
274-4
2-38
315-6
2-43
356-9
2-47
398-1
2-51
489-4
2-54
8-5
284-8
2-41
327-3
2-46
369-8
2-5
412-3
2-54
454-8
2-57
8-75
295-4
2-44
339-1
2-49
382-9
2-53
426-6
2-57
470-4
2-6
9
306
2-46
351
2-52
396
2-56
441
2-6
486
2-62
9-25
316-9
2-49
363-1
2-54
409-4
2-59
455-6
2-62
501-9
2-66
9-5
327-8
2-51
375-3
2-57
422-8
2-61
470-3
2-65
517-8
2-68
9-75
338-9
2-54
387-6
2-6
436-4
2-64
485-1
2-68
533-9
2-71
10
350
2-56
400
2-62
450
2-67
500
2-71
550
2-74
10-5
530-3
2-76
582-8
2-79'
11
...
...
561
2-81
616
2-85
11-5
•••
593-3
2-86
650-8
2-9
12
...
624
2-91
684
2-94
238
HYDRAULICS
Table XLV. — Continued. (1 to 1.)
Depth
of
Water.
Bed 60 feet.
Bed CO feet.
Bed 70 feet.
Bed 80 feet.
Bed 90 feet.
A
VS.
A
VS.
A
VR
A
VR
A
VR
Feet.
1
51
-982
61
-985
71
-987
81
-989
91
•99
1-5
n-25
1-19
92-25
1-2
107-3
1-2
2
104
1-37
124
1-39
144
1-35
164
1-38
184
l-'39
2-25
117-6
1-44
140-1
1-45
162-6
1-46
185-1
1-46
207-6
1-47
2-5
131-3
1-52
156-3
1-53
181-3
1-53
206-3
1-54
231-3
1-54
2-75
145-1
1-58
172-6
1-6
200-1
1-6
2-27-6
1-61
2.55-1
1-61
3
159
1-65
189
1-66
219
1-67
249
1-68
279
1-68
3-25
173-1
1-71
205-6
1-72
238-1
1-73
270-6
1-74
303-1
1-75
3-5
M7-3
1-77
222-3
1-78
257-3
1-79
292-3
1-8
327-3
1-81
3-75
201-6
1-82
239-1
1-84
276-6
1-85
314-1
1-86
351-6
1-87
4
216
1-88
256
1-9
296
1-91
336
1-92
376
1-93
4-25
230-6
1-93
273 1
1-95
315-6
1-96
358-1
1-97
4(X)-6
1-98
4-5
245-3
1-98
290-3
2
335-3
2-01
380-3
2 03
425-3
2-03
4-75
260-1
2-03
307-6
2-05
365-1
2 06
402-6
2-08
450-1
2-09
5
275
2-07
325
2-1
375
2-11
425
2-13
475
2 14
5-25
290-1
2-12
342-6
2-14
395-1
2-16
447-6
2-17
500-1
218
5 '5
305-3
2-16
360-3
2-18
415-3
2-2
470-3
2-2-2
525-3
2-23
5-75
320-6
2-2
378-1
2-23
435-6
2-25
493-1
2-26
550-6
2-28
6
336
2-24
396
2-27
456
2-29
516
2-31
576
2-32
6-25
351-6
2-28
414-1
2-31
476-6
2-33
539-1
2-35
601-6
2-36
6-5
367-3
2-32
433-3
2-35
497-3
2-37
562-3
2-39
6-27-3
2-41
6-75
383-1
2-35
450-6
2-39
518-1
2-41
585-6
2-43
653-1
2-45
7
399
2-39
469
2-42
539
2-45
609
2-47
679
2-49
7-25
415-1
2-43
487-6
2-46
560-1
2-49
632-6
2-51
705-1
2-53
7-5
437-3
2-46
506-3
2-5
581-3
2-52
656-3
2-55
731-3
2-56
7-75
447-6
2-49
525-1
2-53
602-6
2-56
680-1
2-58
757-6
2-6
8
464
2-53
544
2-57
624
2-6
704
2-62
784
2-64
8-25
480-6
2-56
563 1
2-6
645-6
2-63
7-28 1
2-65
810-6
2-67
8-5
497-3
2-59
582-3
2-63
667-3
2-66
752-3
2-69
837-3
2-71
8-75
514-1
2-62
601-6
2-66
689-1
2-7
776-6
2-72
864-1
2-74
9
531
2-65
621
2-7
711
2-73
801
2-76
891
2-78
9-25
548-1
2-68
640-6
2-73
733-1
2-76
S'J.i-li
8-79
918-1
2-81
9-5
565-3
2-71
660-3
2-76
7.'>.~> '',\
2-79
S5II-S
2-82
945-3
2 84
9-75
,'-)S2-6
2-74
680-1
2-79
777 -t)
2-S2
875 1
2-85
972-6
2-88
10
600
2 77
700
2-82
800
2-8.->
900
2-88
1000
2-91
103
635-3
2 82
740-3
2-87
S4.-1-:!
2-91
950-3
2-94 ;i055
2-97
11
671
2-87
781
2-93
891
2-97
1001
3
nil
3-03
11-5
708-3
2-93
823-3
2-98
938-3
3 1)2
1053
3-06
1168
3-09
12
744
2-98
864
3-03
984
3-08
1104
3-11
1224
3-14
OPEN CHANNELS — UNIFORM FLOW
239
Table XhY .—Continued. (1 to 1.)
Depth
of
Water.
Bed 100 feet.
Bed 120 feet.
Bed 140 feet.
Bed 160 feet.
A
VH
A
^K
A
ViJ
A
^R
Feet.
1
101
■991
121
■992
141
■993
161
994
2
2U4
1-39
224
1-39
284
14
324
4
2-25
230 1
1-47
275
1
1-48
320 1
\-il
360-1
48
2 5
2.)G-3
1-55
306
3
1^55
356-3
1 -50
406-3
56
2-75
282-6
1-62
337
6
r63
392-6
163
447-6
64
3
309
1-69
369
1-7
429
1-7
489
7
3-25
335-6
1 -/.■)
400
6
1-76
465-6
r77
530 ■&
77
3-5
362 3
1 S2
432
3
1-82
502 3
1 -xs
572-3
84
3-75
389-1
1-88
464
1
1-89
539-1
1-89
614-1
9
4
416
1 99
496
1-94
576
r95
656
1
96
4-25
443-1
1-09
528
1
2
613-1
2 01
698-1
2
01
4-5
470-3
2-04
560
3
2^05
650-3
2^06
740-3
2
07
4-75
497-6
2-09
592
6
211
687 6
2-12
782-6
2
12
5
525
2-15
625
2-16
725
2^17
825
2
18
5-25
552-6
2-19
657
6
2-21
762-6
2 22
867-6
2
23
55
580-3
2-24
690
3
2-26
800-3
2-27
910-3
2
28
5-75
608-1
2-29
723
1
2-3
838-1
2 -.32
953-1
2
33
6
036
2-33
756
2-35
876
2-36
996
2
37
6-25
664 1
2-38
789
1
2 39
914-1
2-41
1039
■2
41
6-5
692-3
2-42
822
3
2-44
952-3
2-45
1082
2
46
6-75
720-6
2-46
855
6
2-48
990-6
2-5
1126
2
51
7
749
2-5
889
2-52
10-29
2-54
1169
2
55
7-25
777-6
2 54
922
6
2-56
1068
2-58
1213
2
59
7-5
806-3
2-58
O.-iO
3
2-6
1106
£■62
1256
2
63
7-75
835-1
2-62
990
1
2-64
1145
2-66
1300
2
67
8
864
2-65
]0-24
2 68
1184
2^7
1344
2
71
8-25
893-1
2-69
1058
2-72
1-223
2-74
1386
2
75
8-5
922-3
2-73
1092
2-75
1-202
2^77
14.32
2
79
8-75
951-6
2-76
1127
2-79
1302
281
1477
2
83
9
981
2-8
1161
2^83
1341
2^85
1521
2
86
9-25
1011
2-83
1196
2-86
1381
2-88
1506
2
9
9-0
1040
2-86
1230
2^89
1420
2-92
1610
2
94
9-75
1070
2-9
1265
2-93
1460
2 95
1655
2
97
10
1100
2-93
1300
2-96
1.500
2-99
1700
3
01
10-5
1160
2-99
1370
3-03
l.VSO
3-05
1790
3
07
11
1221
3-05
1441
3-09
1661
312
1881
3
14
11-5
1282
3-11
1512
3^15
1642
3-18
1972
3
2
12
1344
3-17
1584
3^21
18-24
3-25
-2064
3-26
240
HYDRAULICS
Table XLVI. — Sectional Data for Open Channels.
Trapezoidal Sections — Side-slopes 1^ to 1.
\
1
Depth
of
Water.
Bed 1 foot.
Bed 2 feet.
Bed 3 feet.
Bed i feet
Bed 5 feet.
A
^/B
A
VB
A
VB
A
VM
A
VR
Feet.
•5
•87
■56
r38
•6
1^88
•63
2-38
•64
2 •875
•64
•76
1-59
•65
2^34
•71
3^09
•73
3-84
•76
4-59
•77
1
2-5
•74
3^5
•79
4^5
•83
5-5
•85
6-5
•87
1-25
3-59
■81
4^84
•86
6 09
•9
7-34
•93
8 59
•95
1-5
4-48
•87
6-37
•93
7^87
•97
9-37
1
1087
102
1-75
6-34
■93
8^09
•99
984
103
11-59
106
13-34
V09
2
8
■99
10
1^04
12
1^08
14
112
16
115
2-25
9-84
ro4
12-09
109
1434
114
16-59
1-17
18^84
1-2
v2-5
11-87
ro9
14-37
1^14
16^87
119
19-37
1-22
21^87
1-25
2-75
1409
1^14
16-84
119
1959
1^23
•2-2 34
1-27
25 09
13
3
16-5
M8
19-50
1-23
22^5
1^28 j -loo
1-31
28 5
134
3-25
22-34
1-28
25 6
1-3-2 28^84
1-36
3209
1.^
3-5
25-37
1-32
28^87
136
3237
1-4
35 87
143
3-75
28^6
1-36
3234
14
36 09
1-44
39 84
1-47
4
...
32
1-39
36
144
40
1-47
44
151
4-2,'5
_
...
39 84
1-48
44 •09
1-51
48 34
1^53
4 5
...
43^87
1-51
4S-S7
1-55
52^87
r58
4-75
...
48 '09
1 -55
52^84
1-58
57 ^59
161
5
.V2-,-)
1 -58
57 •o
1-62
62^5
164
OPEN CHANNELS — UNIFORM FLOW
241
Table XLYL— Continued. (1-| to 1.)
Depth
of
Water.
Bed 6 feet.
Bed 7 feet.
Bed 8 feet.
Bed 9 feet.
Bed 10 feet.
A
ViJ
A
ViJ
A
VJJ
A
VK
A
VJs
Feet.
•5
3-37
-66
3-87
67
4-37
■67
4-88
■68
5-38
-68
•75
5-34
-78
6 09
79
6-84
-8
7-59
-81
8-34
-81
1
7-5
-89
8-5
89
9-5
-9
10-5
-91
11-5
-92
1-25
9-84
-97
11 09
98
12-34
-99
13-59
1
14-84
1-01
1-5
12-37
1-04
13-87
06
15-37
1-07
16-88
1-08
18-38
1-09
1-75
15 -09
111
16-84
12
18-59
1-14
20-34
1-15
22-09
1-16
2
18
1-17
20
18
22
1-2
24
1-22
26
1-23
2-25
21-09
1-23
23-34
24
25-59
1-26
27-84
1-28
30-09
1-29
2-5
24-37
1-28
26-87
3
29-37
1-31
31-88
1-33
34-38
1-34
2-75
27-84
1-33
30-59
35
33-34
1-36
36-09
1-38
38-84
1-39
3
31-5
1-37
34-5
39
37-5
1-41
40-5
1-43
43-5
1-44
3-25
35-34
1-41
38-59
44
41-84
1-46
45-09
1-48
48-34
1-49
3-5
39-37
1-45
42-87
48
46-37
1-5
49-88
1-52
53-38
1-54
3-75
43-59
1-49
47-34
52
51-09
1-54
54-84
1-56
58-59
1-58
4
48
1-53
52
56
56
1-58
60
1-6
64
1-62
4-25
52-59
1-57
56-54
59
6109
1-62
65-34
1-64
69-59
1-66
4-0
57-37
1'6
61-87
63
66-37
1-65
70-88
1-68
75-38
1-7
4-75
62-34
1-64
67 09
66
71-84
1-69
76-59
1-71
81-34
1-74
5
67-5
1-67
72-5
7
77-5
1-72
82-5
1-75
87-5
1-77
5-25
72-84
1-71
78 09
73
88-34
1-76
88-59
1-78
93-84
1-8
5-5
78-37
1-74
83-87
77
89-37
1-79
94-87
1-81
100-4
1-83
5-75
84-09
1-77
89-84
8
95-59
1-83
101-34
1-85
107-1
1-87
6
90
1-81
96
83
102
1-85
108
1-88
114
1-9
6-25
114-8
1-91
121-1
1-93
6-5
121-9
1-94
128-4
1-96
6-75
129-1
1-97
135-9
1-99
7
...
136-5
2
143-5
2-02
7-25
144-1
2-03
151-4
2-05
7-5
...
151-9
2-05
159-4
2-07
7-75
159-8
2-08
167-6
2-1
8
...
168
2-11
176
2-13
242
HYDKAULICS
Table XLVI.-
-Gontmued.
m to 1.)
•
Depth
of
Water.
Bed 12 feet.
Bed 14 feet.
Bed 16 feet.
Bed 18 feet.
Bed 20 feet
A
Vis
A
Vie
A
ViJ
A
VR
A
^R
Feet.
•5
6-37
•68
7-37
-68
8-37
-69
•75
9-84
•82
11-34
■82
12-84
-83
14-34
•'83
I's's
-84
1
13-5
-93
15-5
•93
17-5
•94
19-5
•95
21-5
-95
1-25
17-34
1-02
19-84
1-04
22-34
1^04
24-84
1^05
27-34
1-05
1-5 •
21-38
1-11
24-37
1-12
27-37
1-13
30-37
1-14
33-37
1-15
1-75
25-59
1-18
29-09
1-2
32-59
1-21
36-09
1-22
39-59
1-23
2
30
1-25
34
1-26
38
1-28
42
1-29
46
1-3
2-25
34-59
1-31
39-09
1-33
43 59
1-34
48-09
1-36
52-59
1-37
2-5
39-38
1-37
44-37
1-39
49-37
1-4
54-37
1-42
59-37
1-43
2 '75
44-34
1-42
49-84
1-44
55-34
1-46
60-84
1-48
66-34
1-49
3
49-5
1-47
55-5
1-5
61-5
1-51
67-50
1-53
73-5
1-54
3-25
54-84
1-52
61-34
1-55
67-84
1-56
74-34
1^58
80-84
1-6
3-5
60-38
1-57
67-37
1-59
74-37
1-61
81-37
1-63
88-37
1-65
3-75
66-09
1-61
73-59
1-64
81-09
1-66
88-59
1-68
96-09
1-69
4
72
1-65
80
1-68
88
1-7
96
1-72
104
1-73
4-25
78-09
1-69
86-59
1-72
95-09
1-74
103-6
1-76
1121
1-78
4-5
84-38
1-73
93-37
1-76
102-4
1-78
111-4
1-8
120-4
1-82
4-75
90-84
1-76
100-3
1-79
109-8
1-82
119-3
1-84
128-8
1-86
5
97-5
1-8
107-5
1-83
117-5
1-86
127-5
1-88
137-5
1-9
5 25
104-3
1-83
114-8
1-86
125-3
1-89
135-8
1-91
146-3
1-94
5-5
111-4
1-87
122-4
1-9
133-4
1-93
144-4
1-95
155-4
1-97
5-75
118-6
1-9
130-1
1-93
141-6
1^96
153-1
1-98
164-6
2 01
6
126
1-94
138
1-97
150
2
182
202
174
204
6-25
133-6
1-96
146-1
2
158-6
2-03
171-1
205
183-6
2-08
6-5
141-4
2
154-4
2-03
167-4
2-06
180-4
2-09
193-4
2-11
6-75
149-4
2-02
162-9
2-06
176-4
2-09
189-9
2-12
203-4
2-14
7
157-5
2-05
171-5
2-09
185-5
2-12
199-5
2-15
213-5
2-17
7-25
165-9
2-08
180-4
2-12
194-9
2-15
209-4
218
223-9
2-2
7-5
174-4
2-11
189-5
2-15
204-4
218
219-4
2-21
234-4
2-23
7-75
183-1
2-14
198-6
2-17
2141
2-21
229-6
2-24
245-1
2-26
8
192
2-17
208
2-2
224
2-24
240
2-27
256
2-28
8-25
• *■
...
250-6
2-3
267-1
2-31
8-6
...
...
...
261-4
2-32
278-4
2-34
8-75
272-3
2-34
289-8
2-37
9
2S3-5
2-37
301-5
2-4
9-25
•t.
294-8
2-4
313-3
2-42
9-5
...
306-4
2-42
325-4
2-45
9'75
...
3181
2-45
337-6
2-47
10
330
2-47
360
2-5
OPEN CHANNELS — UNIFORM FLOW
243
Table XLYl.—Coniinued. (1| to 1.)
Bed 26 feet.
Bed 30 feet
Bed 35 feet
Bed 40 feet
Bed 45 feet
Depth
of
Water.
A
ViJ
A
^R
A
ViJ
A
v*
A
^B
Pbet.
1
26-5
-96
31-5
•97
36-5
•97
41-5
•98
46-5
•98
1-5
40-88
1-16
48-38
1-18
55-88
1-18
63-38
1-18
70-88
1-19
2
56
1-32
66
1-33
76
1-34
86
1-35
96
1-36
2-25
63-84
1-39
75-09
1-4
86-34
1-41
97-59
1-42
108-8
1-43
2-5
71-88
1-45
84-37
1-47
96-88
1-48
109-4
1-49
121-9
1-5
2-75
80-09
1-51
93-84
1-53
107-6
1-55
121-3
1-56
135-1
1-57
3
88-5
1-57
103-6
1-59
118-5
1-61
133-5
1-62
148-5
1-63
3-25
87-09
1-63
113-3
1-65
129-6
1-67
145-8
1-68
162-1
1-69
3-5
105-9
1-68
123-4
1-7
140-9
1-72
158-4
1-73
175-9
1-76
3-75
114-8
1-73
133-6
1-75
152-3
1-77
171-1
1-79
189-8
1-8
4
124
1-78
144
1-8
164
1-82
184
1-84
204
1-85
4-25
133-3
1-82
154-6
1-85
175-8
1-87
1971
1-89
218-3
1-9
4-5
142-9
1-86
165-4
1-89
187-9
1-91
210-4
1-93
232-9
1-95
4-75
152-6
1-9
176-3
1-93
200-1
1-96
223-8
1-98
247-6
2
5
162-5
1-94
187-5
1-97
212-5
2
237-5
2-03
262-5
2-04
5-25
172-6
1-98
198-8
2-01
2-25-1
2-04
251-3
2-07
277-6
2-08
5-5
182-9
2-02
210-4
2-05
237-9
2-08
265-4
2-11
292-9
2-13
5-75
193-3
2-06
222
2-09
250-8
2-12
279-6
2-15
308-3
2-16
6
204
2-09
234
2-13
264
2-16
294
2-18
324
2-2
6-25
214-8
2-13
246-1
2-16
277-3
2-2
308-6
2-22
339-8
2-24
6-5
225-9
2-16
258-4
2-2
290-9
2-23
323-4
2-26
356
2-28
6-75
237-1
2-19
270-9
2-23
304-6
2-27
338-4
2-29
372-1
2-32
7
248-5
2-22
283-5
2-27
318-5
2-3
353-5
2-33
388-5
2-35
7-25
260-1
2-25
296-4
2-3
332-6
2-33
368-9
2-36
405-1
2-38
7-5
271-9
2-29
309-4
2-33
346-9
2-36
384-4
2-39
421-9
2-42
7-75
283-8
2-31
322-6
2-36
361-3
2-39
400-1
2-43
438-8
2-45
8
296
2-34
336
2-39
376
2-42
416
2-46
456
2-48
8-25
308-4
2-37
349-6
2-42
390-9
2-45
432-1
2-49
473-4
2-51
8-5
320-9
2-4
363-4
2-45
405-9
2-48
448-4
2-52
490-9
2-55
8-75
333-6
2-43
377-3
2-48
421-1
2-51
464-8
2-55
508-6
2-58
9
346-5
2-46
391-5
2-5
436-5
2-54
481-5
2 -.58
526-5
2-61
9-25
359-6
2-48
405-8
2-53
452-1
2-57
498-3
2-61
544-6
2-64
9-5
372-9
2-51
420-4
2-56
467-9
2-6
515-4
2-64
562-9
2-66
9-75
386-4
2.53
435-1
2-58
483-9
2-63
532-5
2-66
581-3
2-69
10
400
2-56
450
2-61
500
2-65
550
2-69
600
2-72
10-5
585-4
2-74
637-9
2-77
11
...
621-5
2-79
676-5
2-83
11-5
...
658-4
2-84
715-9
2-83
12
...
...
696
2-89
756
2-93
244
HYDRAULICS
Table XlNl.— Continued. (IJ to 1.)
Depth
of
Water
Bed 60 feet.
Bed 60 feet.
Bed 70 feet.
Bed 80 feet.
Bed 90 feet.
A
VJJ
A
^n
A
^R
A
^R
A
^R
Feet.
1
51 -5
-98
61-5
-98
71-5
-98
81-5
...
91-6
1-5
78-38
1-19
91-13
1-18
108-4
1-19
123-4
138-4
2
106
1-36
126
1-37
146
1-37
166
186
..."
2-25
120-1
1-44
142-6
1-46
165-1
1-46
187-6
210-1
2-5
134-4
1-51
159-4
1-52
184-4
1-53
209-4
234-4
B-75
148-8
1-58
176-3
1-59
203-8
1-6
231-3
258-8
3
163-5
1-64
193-5
1-65
223-5
1-66
253-6
283-5
3-25
178-3
1-7
210-8
1-71
243-3
1-73
275-8
308-3
3-5
193-4
1-76
228-4
1-77
263-4
1-79
298-4
333-4
3-75
208-6
1-81
246-1
1-83
283-6
1-84
321-1
...
358-6
4
224
1-86
264
1-88
304
1-9
344
384
4-25
239-6
1-92
282-1
1-94
324-6
1-95
367-1
409-6
4-5
255-4
1-96
300-4
1-99
345-4
2
390-4
...
435-4
4 '75
271-3
2-01
318-8
2-03
366-3
205
413-8
...
461-3
5
287-5
2-05
337-5
2-08
387-6
2-1
437-5
..■
487-5
5-25
303-8
2-1
356-3
2-12
408-8
2-14
461-3
513-8
5-5
320-4
2-14
375-4
2-17
430-4
2-19
485-4
540-4
5-75
337-1
2-18
394-6
2-21
452-1
2-23
509-6
567-1
6
354
2-22
414
2-25
474
2-27
634
694
6-25
371-1
2-26
433-6
2-29
496-1
2-32
658-6
...
621-1
6-5
388-4
2-3
453-4
2-33
518-4
2-36
583-4
648-4
6-75
405-9
473-4
540-9
...
608-4
...
675-9
...
7
423-5
2-37
493-5
2-4
563-6
2-43
633-6
703-6
7-25
441-4
(>•
513-9
686-4
• ■■
668-9
731-4
7-5
469-4
2-44
534-4
2-47
609-4
2-51
684-4
759-4
7-75
477-6
2-47
555-1
632-6
2-54
710-1
787-6
8
496
2-5
576
2-54
656
2-67
736
...
816
8-25
514-6
597-1
...
679-6
762-1
844-6
8-0
533-4
2-67
618-4
2-61
703-4
2-64
788-4
873-4
...
8-75
552-3
2-6
639-8
2-64
727-3
814-8
902-3
9
571-6
2-63
661-5
2-67
757-6
2-71
841-6
931-5
9-25
590-8
683-3
2-7
775-8
2-74
868-3
960-8
9'5
610-4
2-69
705-4
2-73
800-4
2-77
895-4
...
990-4
9-75
630
2-72
727-6
2-76
826
2-8
922-5
1020
10
650
2-75
750
2-79
850
2-83
960
1050
...
10-5
690-4
2-8
795-4
2-85
900-4
2-89
1005
"'. 1110
11
731-6
2-86
841-0
2-9
961-5
2-94
1062
...
1172
11-5
773-4
2-91
888-4
2-96
1008
3
11,18
...
1233
12
816
2-96
936
3-01
1066
3-05
1176
1296
OPEN CHANNELS — UNIFORM FLOW
245
Table XL VII.— Sectional Data for Oval Sewers. (Art. 3.)
Metropolitan Ovoid.
Dimensions.
Full.
Two-thirds full.
One-third full.
^
.JR
A
v«
A
VJJ
1' 0" X 1' 6"
1-15
-54
•76
-56
-28
•45
1' 2"xl' 9"
116
•58
103
-61
-39
■49
• r 4" X 2' 0"
2-04
•62
134
-65
-51
•53
1' 6"x2' 3"
2-58
•66
1^7
•69
-64
-56
r 8"x2' 6"
319
-69
2-1
•73
-79
•59
l'10"x2' 9"
3-86
•73
2-54
•76
-96
•62
2' 0"x3' 0"
4-59
-76
3-02
-79
1-14
•64
2' 2" X 3' 3"
5-39
-79
3-55
-83
1-33
•67
2' 4" X 3' 6"
6-25
•82
4-12
•86
1-55
-69
2' 6" X 3' 9"
7-18
•85
4-72
•88
1-78
•72
2' 8" X 4' 0"
8-17
•88
5-38
•92
2-02
•74
2' 10" X 4' 3"
9-22
•91
6-07
•95
2-28
•76
3' 0" X 4; 6"
10-34
•93
6-8
•97
2 ■56'
•79
3' 2" X 4' 9"
11-52
•96
7-58
1
2-85
•81
3' 4" X 5' 0"
12-76
•98
8-4
103
3-16
•83
3' 6" X 5' 3"
14-07
1-01
9-26
1-05
3-48
•85
3' 8" X 5' 6"
15-41
1-03
10-16
1-08
3-82
•87
3' 10" X 5' 9"
16-88
1-06
11-11
1-1
4-17
•89
4' 0" X 6' 0"
18-38
1-08
12-09
1-12
4-54
•91
4' 2" X 6' 3"
19-94
1-1
13-12
1-15
4-93
■93
4' 4" X 6' 6"
21-57
1-12
1419
1-17
5-33
■95
4' 6" X 6' 9"
23-26
114
15-31
1-19
5-75
•96
4' 8" X 7' 0"
25-01
1-16
16-46
1-21
6-19
•98
4' 10" X 7' 3"
26-83
1-18
17-66
1-23
6-64
1
5' 0" X 7' 6"
28-71
1-2
18-9
1-26
7-1
r02
5' 2" X r 9"
30-67
1-22
20^18
1-28
7-58
ro3
5' 4" X 8' 0"
32-67
1-24
21-5
1-3
8-08
ro5
5'. 6" X 8' 3"
34-74
1-26
22-86
1-32
•8^59
i-vn
246
HYDRAULICS
Table XL VIII. — Sectional Data eor Oval Sewers. (Art. 3.)
Hawhsley's Ovoid.
Pul
.
Two-thirds fuU.
One-third full.
Transverse
Diameter,
^
ViJ
A
VB
A
VJJ
V 0"
1
-53
■67
-56
-26
-44
1' 2"
1-36
-57
-91
•6
•35
-48.
r 4"
1-77
-61
1-19
-64
•46
-51
1' 6"
2-24
-64
1-51
-68
•58
-54
1' 8"
2-77
-68
•71 ^
1-87
-72
■71
■57
r 10"
2-35
2-25
•75
•86
•6
2' 0"
3-98
'74
2-69
-79
103
-63
2' 2"
4-67
•77
314
-82
1-21
-66
2' 4"
5-42
•8
3-66
■85
1-4
•68
2' 6"
6-2^
-83
4-2
-88
1-61
"7
2' 8"
7-08
■86
4-77
-91
1-83
-72
2' 10"
7-89
•89
5 38
-94
- 2-06
•74
3' 0"
8-97
-91
6 04
-96
2-31
•77
3' 2"
.9-9fe
•94
6-73
-99
2-58
•79
3' 4"
11-06
-96
7-46
1 -ia
2-85
•81
3' 6"
12-2
-98
8-22
104
315
•83
3' 8"
13-38
1-01
9
107
3-45
•85
3' 10"
14-63
103
9-87
1-09
3-78
•87
4' 0"
15-93
1-05
10-74
1-11
4-11
•89
4' 2"
17-28
1-07
11 66
1-14
4-46
•91
4' 4"
18-69
109
12-57
1-16
4-82
•93
4' 6"
20-18
1-12
13-6
1-18
5-20
•94
4' 8"
21-68
1-14
14-62
1-2
5-59
•96
4' 10"
23-25
1-16
15-68
1-22
6
•98
5' 0"
24-89
M8
16-79
1-24
4-2
1
5' 2"
26-57
1-2
17-92
1-27
6^86
101
5' 4"
28-32
1-21
191
1-29
7-31
1-03
5' 6"
3011
1-23
20-26
1-31
7-76
1-04
5' 8"
31-56
1-25
51-5
1-33
S-->4
1-06
5' 10"
33-87
1-27
22-84
1-34
S-74
1-07
6 0"
35-84
1-29
2417
i-:?6
'.1-25
109
OVAL SEWERS — UNIFORM FLOW
247
Table XLIX. — Sectional Data for Oval Sewers. (Art. 3.)
Jackson's Peg-top Section.
Pull
Two-thirds full.
One-third full.
Dimensions.
A
^R
A
ViJ
A
Vi2
1' 0"xl' 6"
1039
•52
•646
•53
-242
-44
1' 2"xl' 9"
1-414
-56
•88
•57
-33
-47
1' 4" X 2' 0"
1-846
-6 .
M48
•61
•431
-5
r 6"x2' 3"
2-337
-63
r453
•65
•545
-53
1' 8" X 2' 6"
2-885
•67
1-793
•68
•65
-56
1' 10" X 2' 9"
3-491
■7
2-115
•72
•813
-59
2' 0"x3' 0"
4-154
•73
2^583
•75
•969
•62
2' 2"x3' 3"
4-874
•76
3^032
•78
M36
•64
2' 4"x3' 6"
5-654
-79
3-516
•81
r319
•67
2' 6"x3' 9"
6-491
-82
4-034
•84
1513
•69
2' 8" X 4' 0"
7-385
•84
4-593
•86
1-722
•71
2'10"x4' 3"
8-337
•87
5-184
•89
1-943
•73
3' 0"x4' 6"
9-347
•89
5-813
-92
2-179
•76
3' 2"x4' 9"
10-41
•92
6-478
-94
2-427
•78
3' 4"x5' 0"
11-54
•94
7-172
-97
2-602
•8
3' 6"x5' 3"
12-72
•97
7-912
-99
2-967
•82
3' 8" X 5' 6"
13-96
•99
8-461
1-01
3-254
•84
3'10"x5' 9"
15-26
roi
9-492
1-03
3-556
•85
4' 0"x6' 0"
16-62
103
10-33
1-06
3-874
•87
4' 2" X 6' 3"
18-03
r06
11-22
1-08
4-201
•89
4' 4" X 6' 6"
19-5
1-08
12-13
1-1
4-542
•91
4' 6"x6' 9"
21-03
1-1
13-08
1-12
4-903
•93
4' 8"x7' 0"
22-62
1-12
14-07
1-14
5-274
■94
4' 10" X 7' 3"
24-26
1-14
15-09
1-16
6-653
•96
5' 0"x7' 6"
25-96
1-16
16-14
M8
6-054
•98
5' 2"x7' 9"
27-72
1-18
17-24
1^2
6-46
•99
5' 4"x8' 0"
29-54
119
18-37
1-22
6-844
1-01
5' 6"x8' 3"
31-42
1-21
19-54
1-24
7-321
1-02
5' 8"x8' 6"
.33-35
\-2Z
20-74
1-26
7-77
1-04
5' 10" X 8' 9"
35.34
1^25
21-98
1-28
8-234
105
6' 0" X 9' 0"
37-39
1-27
23-25
1-3
8-718
1-07
248
HYDRAULICS
Table L. — Ratios of Combined Length of Two
Side-slopes to Depth of Water.
Side-slope =4 to I ftol 1 to 1 IJtol litol 2tol 2i to 1 3tol
Ratio =2-236 25 2'828 3-33§ 3606 4-472 5-385 6-325
These ratios can be used for calculating B for channels outside the range
of tables xliii.-xlvi.
Table La. — Circular Channels' partly full.
(Art. 6).
The Diameter of the Channel is supposed to he \,
Depth of
Water.
Angle sub-
tended by
Wet Portion
of Border.
Relative
Values of .4
Relative
Values of ^R
Feet.
-25
•5
-75
1
120°
180°
240°
360°
•196
•5
-804
1
■767
1
1-1
1
For actual values of A and ^R see table xxiii. , page 142.
CHAPTEE VII
OPEN CHANNELS— VARIABLE FLOW
[For preliminary information see chapter ii. articles 10 to 14 and 17 to 21]
Section I. — Bends and Abrupt Changes
1. Bends. — The loss of head at a change in direction in an open
stream is, as in the case of a pipe, greater for an elbow than for
a bend. The formula for loss of head at a bend arrived at by
observations on the Mississippi is H= — -— - — where is the
angle subtended by the bend. This takes no account of the
radius. In a bend of 90° the loss of head by this formula is
■48——. Generally a single bend with ordinary velocities causes
little heading-np, but if a stream has a long succession of bends
their cumulative effect may be considerable. It is practically the
same as that of an increase of roughness, and may be allowed for
by taking a lower value of the co-efficient C. How far the loss
of head at a bend depends on the radius of the bend is not known.
(Of. chap. V. art. 4.)
At a bend there is a ' set of the stream ' towards the concave
bank, the greatest velocity being near that bank; and there is a
raising of the water-level there, so that
the surface has a transverse slope (Fig.
117). There is also a deepening near
the concave bank and a shoaling at the
opposite one, but this is not all due to
the direct action of centrifugal force.
The high-water level at the concave
bank, due to centrifugal force, gives a greater pressure and
tends to cause a transverse current from the concave towards
the convex bank. This tendency is, in the greater part of the
cross-section, resisted by the centrifugal force. But the -water
near the bed and sides has a low velocity, the centrifugal
•m
250 HYDRAULICS
force is therefore smaller, and transverse flow occurs. Solid
material is thus rolled towards the convex bank, and it accumu-
lates there because the velocity is low. To compensate for the
low-level current towards the convex bank there are high-level
currents towards the concave bank.
The directions of the currents are
shown by the arrows on Fig. 117. In
Fig. 118 the dotted line shows the
direction of the strongest surface cur-
rent and the arrows the currents near
the bed. This explanation is due to
Thomson, and has been confirmed by
him experimentally. When the channel
is of masonry or even very hard soil
the deepening TF^^cannot occur, but
the bank RST may still be formed, the material for it being
brought down by the stream. The greatest velocity is still on the
side next the concave bank.
As the transverse current and transverse surface-slope cannot
commence or end abruptly there is a certain length in which they
vary. In this length the radius of curvature of the bend and the
form of the cross-section also tend to vary. This can often be
seen in plans of river-bends, the curvature being less sharp
towards the ends. This principle has been adopted in construct-
ing river training-walls, and it appears to be sound as tending
against any abruptness in the change of section. For training-
walls to remove bars at the mouth of the Mississippi it has been
proposed to construct, instead of two walls, only one wall having
a curve concave to the stream. The success of this plan would
appear to depend on whether the curve is sharp enough to ensure
the stream keeping close to the wall and not going off in another
direction.
The sectional area of a stream may be less at a bend than in
straight reaches, especially when the channel is hard, so that the
stream cannot excavate a hollow to compensate for the silt-bank;
but the surface-width is often greatest at bends, and in construct-
ing training-walls the width between the walls is sometimes
increased at bonds. In the silt clearances of some tortuous
canals in India it was once the custom to remove the silt RST,
the dotted lino sliowing the section of the cleared channel in the
straight reaches. No allowance was made for the hollow TVW.
A silt-bank so removed quickly forms again. Its removal is
equivalent to the digging of a hole or recess in the bed.
OPEN CHANNELS — VARIABLE FLOW
251
When once a stream has assumed a curved form, be it ever so
slight, the tendency is for the bend to increase. The greater
velocity and greater depth near the concave bank react on each
other, each inducing the other. The concave bank is undermined,
becomes vertical owing to scour of the bed, cracks, falls in, and
is washed away. The bend may go on increasing as indicated by
the dotted lines in Fig. 119, a deposit of silt occurring at the
convex bank, so that the width of the stream
remains tolerably constant. Some of the
large Indian rivers flowing through alluvial
soil sometimes cut away, at bends, hundreds
of acres of land, together with the trees,
crops, and villages standing thereon. Works
to check the erosion would cost many times
as much as the value of the property to be
saved. When a bend has formed in a
channel previously straight, the stream at
the lower end of the bend, by setting against
the bank, tends to cause another bend of the
opposite kind to the first. Thus the ten-
dency is for the stream to become tortuous,
and while the tortuosity is slight the length,
and therefore the slope and velocity, are
little afiected ; but the action may continue
until the increase in the length of the stream
materially flattens the slope, and the conse-
quent reduction in velocity causes erosion to
cease. Or the stream during a flood may
find, along the chord of a bend, a direct
route, with of course a steeper slope. Scouring a channel along this
route it straightens itself, and its action then commences afresh.
2. Changes of Section. — An ' obstruction ' is anything causing
an abrupt decrease of area in a part of the cross-section of a
stream such as a pier or spur. There may or may not be a
decrease in the sectional area of the stream as a whole. There
is a tendency to scour alongside an obstruction owing to the
increased velocity, and downstream of it owing to the eddies.
When a spur is constructed for the purpose of deflecting a stream
or checking erosion of the bank, the scour near the end of the
spur may be very severe, even though there may be very little
contraction of the stream as a whole. If the bed is soft the
spur may be undermined. A continuous lining of the bank with
Fio. 119.
252 HYDRAULICS
protective material is not open to such an attack. Similarly a hole
may be formed alongside of and downstream of a bridge-pier.
The hole may work back to the upstream side of the obstruction,
though there is little original tendency to scour there.
When an obstruction reaches up to the surface, or nearly up to
it, there is a heaping-up of the water on its upstream side due
to the checking of the velocity. In the eddy downstream of an
obstruction the water-level is depressed. The changes of water-
level and velocity are local; that is, they do not necessarily
extend across the stream, and they are independent of the effects
of any general change — supposing such to occur — in the sectional
area of the stream. Their amounts cannot be calculated, but they
often have to be recognised. They should be avoided in observing
water-levels where accuracy is required, as for instance when finding
the surface-slope. The discharge of a branch will be increased by
a spur or obstruction just below it, and decreased by one just
above it. On some irrigation canals in India, where the velocity
is high and the channel of boulders, the cultivators sometimes run
out small spurs below their water-course heads in order to obtain
more water.
An obstruction causes a ' set of the stream,' that is a strong
current, as shown by the arrows in Fig. 119 ; but the distance to
which such a current extends depends entirely on its impetus,
and is not usually great.l If a spur is merely intended to cause
• slack water or silt deposit on its own side of the stream several
short spurs will do as well as one long one, but when the object
is to cause a stream to set against the opposite bank the spurs
may have to be very long.
In a short deep recess in the bed or bank of a stream or down-
stream of an obstruction, if it is large enough to cause dead water,
there is generally a rapid deposit of silt, but not where strong
eddies occur.
When an obstruction causes material reduction of the section of
the stream the velocity past it is increased, and the scour may be
excessive, both from the high velocity past it and (if there is a
subsequent expansion of the stream) the eddies downstream of it.
Thus a partly formed dam EF (Fig. 119) is, unless the gap is
quickly closed, liable to be destroyed by the stream, and so is any
structure which reduces the water-way. In order to lessen scour
of the banks downstream of contracted water-ways the channel
is sometimes widened out so as to form a basin in which the eddies
exhaust themselves.
> See Not63 at end of chapter.
OPEN CHANNELS — VARIABLE FLOW 253
3. Bifurcations and Junctions. — The general effects of these
have been stated in chapter ii. (art. 20). In an irrigation dis-
tributary constructed in India the velocity was exceptionally high,
and it was found that the discharges of some narrow masonry
outlets, taking off from the distributary at right angles, were so
small that it became necessary to rebuild them at a smaller angle.
On the other hand, it was once the custom to build the heads of
the distributaries themselves at an angle of 45° with the canal,
but they are now built at right angles. The velocity in the canal
is 2 or 3 feet per second, and that in the distributary less. A
slight fall into the distributary is not objectionable. A skew
head is suitable in cases where loss of head is not permissible.
When there is a bend in the main stream importance is some-
times attached to the set of the stream as affecting the supply in
a branch taking off on the concave bank. The velocity in the
branch is that due to its slope and to the depth of water in it.
The advantage possessed by the branch as compared with one on
the opposite bank is the greater depth of water, owing to velocity
of approach. This advantage is small except in the case of a
sharp bend and a high velocity.^ A river about 20 feet deep
was eroding the concave bank at a bend. An attempt was made
to divert it by a straight cut, about a mile long, across the bend.
Owing to the high level of the sub-soil water, the cut could only
be dug down to about 2 feet below the water-level of the river.
The slope of the cut was about one-and-a-half times that of the
river, but owing to the small depth of water the velocity was low,
and the cut, or at least its upper part, rapidly silted up. The
reason given for its failure was that its head was not so placed as
to catch the set of the stream at the bend next above. This set
might have given an inch or two more water, and the cut might
have taken a few days longRjr to silt up.
In river diversion works spurs are sometimes used to ' drive
the river ' down a branch channel. A spur may make the current
set against the branch head (art. 2), but unless the spur is so long
' There is also the advantage — very slight unless the velonity is high — due tc
the higher water level at the concave bank.
254
HYDRAULICS
as to greatly contract the water-way, the rise of water-level will
not be great except in cases of very high velocities, and the river
will continue to distribute itself according to the discharging
capacities of the two branches. It is only by closing or thoroughly
obstructing one branch or enlarging the other that the stream can
be forced to alter its distribution of discharge.
At a junction of one stream with another there are the usual
eddies and inequalities in the water-level, all depending as before
on the sharpness of the angle and on the velocity. When the main
stream is not much larger than the tributary, the latter may cause
a set of the current against the opposite bank and erode it.
4. Kelative Velocities in Cross-section. — In every case of
abrupt contraction in a stream there are (chap. ii. art. 21) eddies
which extend back to the point where the fall in the surface
begins. Upstream of these eddies the distribution of the velocities
in the cross-section is not affected. In the case of a pier, even
a wide one, in the middle of a straight uniform stream, the
maximum velocity remains in mid-stream till just before the pier
is reached. If a plank or gate obstructs the upper portion of a
stream from side to side, the surface velocities are affected for
only a short distance upstream. A spur or sudden decrease of
width causes slack water for only a short distance. In all these
cases the state of the flow further upstream, as far as regards the
distribution of the velocities, is precisely the same as if no
obstruction existed. In the case of a weir visual evidence is
wanting, but by analogy the same law holds good.
Section II. — Variable Flow in a Uniform Channel
{General Description)
5. Breaks in Uniformity. — Variable flow may be caused by a
change in slope (Figs. 16 and 17, pp. 24 and 25) or in roughness (Figs.
120 and 121), by a de-
bouchure into a pond
or river (Figs. 122 and
1 23), by a weir (Figs.
124 and 125), by a
change in width (Figs.
126 and 127), or in
bed-level (Figs. 128
^'°-i20. and 129). Heading-
up may be caused by a local contraction or submerged weir
OPEN CHANNELS — VARIABLE FLOW
255
(Fig. 130), but the analogous case of a local enlargement has no
effect. A change of hydraulic radius seldom occurs without a
change of sectional area,
and it need not therefore
be considered as a sepa-
rate case. A bend gener-
ally causes some degree
of heading-up.' In each
case the line BC is the
'natural water-surface ' of
the upper reach, that is,
the surface as it would
have been if no change had occurred. The profiles of the water-
surface touch the natural surface at points far upstream. Above
Fig. 121.
Fia. 124.
It may be slight or even inappreciable.
256
HYDRAULICS
PlO. 126.
these points the flow is uniform if the reach extends far enough.
In heading-up there is a tendency to silt, and in drawing-down
to scour.
In the cases shown in
Figs. 126 to 130 there are
abrupt changes in the sec-
tional area, falls in the sur-
face when the area de-
creases, and perhaps rises
where it increases (chap. ii.
arts. 18 and 19). In Figs.
124 and 125 the weir for-
mula gives the discharge
having reference to the sur-
face above the local fall, which therefore need not be considered.
In the other cases there are no abrupt changes in section, and
therefore no local -_ x
changes in level.
A change of one
kind may be com-
bined with another
so that the change
of water-level is
altered or suppres-
sed. For instance,
the changes of roughness may be accompanied by changes in
slope, so that the water-level in the lower reach is at C and the
s flow is uniform, but any
local falls or rises due to
abrupt change of section
(Figs. 126 to 130) will
remain. The rises are
generally, however, negli-
gible, and the falls are
much reduced if the
changes are not actually
sudden (chap. ii. art. 21).
In all cases, whatever, the upstream level has to accommodate itself
to the downstream level. The water-level in the lower reach or
pond or on the crest of the fall is known or can be ascertained. The
local fall or rise, if any, must be found, and there will be heading-
up or drawing-down or neither in the reach above, according as
Flo. 126.
Fia. 127.
OPEN CHANNELS — VARIABLE FLOW
257
the level found is above or below or equal to tlie natural level in
that reach. ^
When the variable flow extends upstream to a point where
there is another
break in uniform-
ity the flow in the
reach is said to be
' variable through-
out.' If the bed of
the reach is level,
or slopes upward
(Figs. 135and 136, ""'■"°-
p. 240), the flow must be variable throughout, however long the
reach may be and
the surface convex
upward.
In a uniform chan-
nel let CD (Fig. 131)
be a ' flume ' of the
same section as the
rest of the channel,
Fig. 129. but of smoother ma-
terial. If the flume
extended upstream far enough the water-surface would be CGH.
Actually it will be
CGL, GL being a
curve of drawing-
down. The height
DG will generally
be very small, and
no appreciable
change in the velo-
city will be caused,
but if surface-slope observations are made a serious error may
occur if the upstream point of obser-
vation falls at M. The slope required
is ECDL, that actually observed is EM.
Often a flume has vertical sides, and is
of a different section to the rest of the
channel. If the change is made grad-
ually there may possibly be no inter-
ference with the straight line of the water-surface, the smaller
^ See Appendix D.
B
Fig. ISO.
Fig. 131.
258 HYDEAULICS
sectional area and hydraulic radius of the flume compensating for
its smoother material. But this is not likely to be the case exactly.
If the change of section is abrupt there will be a change in the
water-level at the entrance of the flume. In the Roorkee Hydraulic
Experiments observations were made in a masonry aqueduct
900 feet long in the Ganges Canal. The surface-slope, instead of
being observed within the aqueduct, was obtained from points
lying far outside it in the earthen channel, and the results of the
experiments, so far as concerns the relation between slope and
velocity in masonry channels, were vitiated.^
6. Bifurcations and Junctions. — A bifurcation or junction may
cause variable flow upstream of it. At a junction let Qi and Q^
be the discharges of the two tributaries. The flow in the main
stream is uniform, and its water-level is that corresponding to the
discharge <3i + Oa. If the conditions of the debouchure of either
tributary are such as to cause any local fall or rise, the amount of
this must be estimated, and the water-level in the tributary just
above the junction is then known. There will be heading-up or
drawing-down or neither in the tributary, according as its natural
water-level is below or above or equal to that so found. There may
be heading-up in one tributary and drawing-down in the other.
At a bifurcation let Q be the discharge of the main stream.
The flow in the branches is uniform. Assume discharges Qi and
02 for them — Qi + Q^ being equal to Q — and find their water-
levels. Allow for any local, fall or rise, and if the water-levels
upstream of them are equal the assumed discharges d and Q. are
correct, and the water-level found is that of the main stream. If
they are not equal it is necessary to alter the quantities Q^ and Q.,
and make a second trial. In the main stream there will be head-
ing-up or drawing-down or neither, according as the water-level
found is higher or lower than, or equal to, its natural water-level.
If a stream flows out of a reservoir the flow will be uniform down-
stream of the fall in the surface (chap. iv. art. 15) which occurs at
the head. If more than two streams meet or separate at one place
the discharges Qi, Q^, Q,, etc., must be considered, and the above
processes adopted. The variable flow caused by a junction or
bifurcation may be counteracted wholly or partly by any other
cause, just as in the other instances of variable flow.
In a jiaper - on the designing of trapezoidal notches at canal falls
it has been observed that a distributary usually takes ofi' a short
' Trmwaclions, i^ocicty of Engineers, 1886.
^ Punjab Irrigation Branch Pajter, No. 2.
OPEN CHANNELS — VARIABLE TLOW 259
distance above a fall, and that though the notch must obviously
be able to pass the whole discharge when the distributary is
closed, it has to be settled in each case whether the design of the
notch should be such as to cause draw when the distributary is
open or heading-up when it is closed. The question must occur
with every distributary, and not only with those taking off above
falls. If the canal is designed so as to give uniform flow with
the distributary closed, then there must be draw when it is open.
If there is uniform flow when the distributary is open, there must
be heading-up when it is closed. The best arrangement depends
on engineering considerations which need not be discussed here.
The opening of an escape or branch may cause scouring up-
stream of it. One method of freeing the upper reach of a canal
from silt is to make an escape from a point some distance below
its head leading back to the river. If there is a weir across the
river the slope of the escape may be great. By opening the
escape scour is caused in the canal, but this may cause some
deposit in the canal downstream of the escape, unless it can be
shut off when the escape is opened.
There were once to be seen in a large canal two gauges, one
just above and the other just below the off-take of an escape
channel. It was stated that the two gauges had been erected in
order that, by noting the difference of their readings, the quantity
of water passing down the escape could be estimated. Both
gauges were carefully read, and copies of the readings sent to
various officials. But when the escape was opened the water-level
on the upper gauge fell practically as much as that on the lower
one. Both gauges always read the same. The assistant in charge
put up a temporary gauge half a mile upstream. This also fell
when the escape was opened. The proper arrangement in such a
case is to have one gauge in the canal below the escape and one
in the escape. Again, some irrigators who wanted a new water-
course were anxious that its off-take should be placed just above
and not just .below the off-take of an existing water-course.
Practically it made no difference whether it was above or below.
There was no sudden fall in the water-level of the canal. If a
branch whose discharge is to be q is to be supplied from a channel
whose discharge is Q, it is necessary first to find what the water-
level in the channel will be when its discharge is Q—q, and then
to design the branch so that it will obtain a discharge q with the
water-level thus found.
7. Effect of Change in the Discharge. — An increase or decrease of
260 HYDRAULICS
the discharge is always accompanied by a rise or fall of the water
level throughout every reach except at the points A (Figs. 122
and 123), where the stream enters or leaves a river or pond whose
water-level is not affected by the alteration of discharge. It is
clear, however, that for a given change of discharge the changes in
the water-levels at two distant points may be very diflferent from
one another. In changes of slope, roughness, width, or bed-level,
a change in the discharge causes no change in the character of the
flow, that is, there is always heading-iip or draw, whichever there
was at first. In a local contraction there is always heading-up,
and also with a weir, except when deeply drowned, if there is no
fall in the bed. In the other cases (debouchures or weirs with
falls) there will be heading-up if the supply falls low enough, and
drawing-down if it rises high enough.
At a bifurcation, if the branches are such that the flow in the
main stream is uniform with the average discharge, and if the
beds of all three channels are at one level, the flow in the main
stream will probably be nearly uniform with all discharges. At
a junction a similar rule obtains only if the discharges of the
tributaries vary in the same proportion.
Above a weir or a rise in the bed the water approaches the line
BE (Figs 124 and 128) as the discharge is reduced, the tendency
to silt increases, supposing the water to be silt-laden, and deposit
will doubtless occur if the discharge falls low enough. A fall in
the bed (Fig. 129) is converted into a clear 'fall' (Fig. 79, p. 99)
at low supply, and in that case there will probably be scour or
' cutting back ' owing to the high velocity.
8. Effects of Alterations in a Channel. — When a natural or
artificial change occurs in a channel, such as deepening, widening,
silting, the erection or removal of a structure, or the manipulation
of a gate or sluice, the consequent change of water-level may
extend upstream to a bifurcation and so affect the discharge. If
the bifurcation is from a body of water whose level is not afiected,
the depth at the head of the channel remains constant, but the
surface-slope alters, and with it the discharge ; or a change in
the channel may cause an altei'ation in the quantity of water lost
by evaporation, percolation, or flooding, and so affect the discharge
But if the discharge of the channel is unaltered, the effect on the
water-level and velocity caused by any change in the channel is
wholly upstream. The building, for instance, of a weir in a stream
ordinarily causes little difference to persons further down the
stream as long as water is not permanently diverted.
OPEN CHANNELS — VARIABLE FLOW 261
In a discussion ^ on some oblique weirs erected in the Severn
it is implied that the weirs caused a lowering of the flood-level
and a deepening upstream. Above the weirs basins had been
made by widening the channel, and the widening might, by itself,
have caused some slight reduction in the flood-level, but not when
a weir was added. It was not contended that the flood discharge
at the weir was reduced. The water-level at D (Fig. 130) would
therefore be the same as it was originally, and since there must
always be some fall from A to D, the flood-level at A must have
been raised. No deepening due to the weir could occur except
close alongside a very oblique weir. (See also chap. iv. art. 18.)
Upstream of a place where changes occur a gauge-reading
affords no proper indication of the discharge, and a discharge
table, if it can be made at all, must be one of double entry, show-
ing the discharge as depending not only on the gauge-reading, but
on other conditions. If gates or shutters are worked there may
be any number of water-surfaces corresponding to one discharge.
An instance of this has already been ^iven in the case of the flow
upstream of an escape. Gauges are sometimes fixed in canals
near their heads, and tables are made showing the discharges as
depending on the gauge-reading. The deposit of silt in the heads
alters the discharges, vitiates the tables, and destroys the utility
of statistics based on the discharges obtained from them. Gauges
ought to be placed below the reaches in which the deposits occur.
The deposit of silt changes both the section and the slope, and
it is next to impossible to allow for it by merely observing the
depth at the gauge.
Sometimes masses of silt are said to travel down a stream. On the
Western Jumna Canal there is a gauge at Jhind and another about
twenty miles upstream. When the upper gauge is kept steady that
at Jhind sometimes slowly rises, although no water is introduced in
the intervening reaches. This has been ascribed to travelling masses
of silt. What happens is that there is scour downstream of the
upper gauge or silting downstream of the lower gauge, or both.
If a channel AB (Fig. 119) is drawn from a source whose water-
level is not afi'ected, and if, near the head of the channel, a branch
BC is taken off', the discharge of the channel below B may be
very little afi'ected. A very slight lowering of the water-level at
B increases the slope AB, and causes more water to be drawn in.
The water-level in the channel may rise slightly at B (chap. ii.
art. 20). A case occurred in which an engineer, wishing to
1 Minutes of Proceedings, Institution of Civil Engineers, vol. Ix.
262 HYDRAULICS
reduce the supply in an overcharged canal, caused a breach to be
made in the bank a short distance below its ofF-take from the
river. He was surprised to find, that although a large volume of
water passed out of the breach, there was no appreciable diminu-
tion of the canal discharge below the breach. In the case of an
irrigation distributary which takes out of a canal, and has itself a
number of water-courses taking out of it not far from its head,
the discharge of the distributary may partly depend on whether
the water-courses are open or not. (Cf. case of branched water-
main, chap. V. art. 3.)
Let a straight cut be made across a bend in a uniform stream.
The slope in the cut is increased and the longitudinal section is as
in Fig. 132. If the discharge is
unaltered the water-level at B
is as before, and there is ten-
dency to scour at A and to silt
at B. The bed and water-
j.,0 132. . surface tend to assume the
positions shown by the dotted
lines, and the probability of this occurring must be considered
in making a cut. If it is desired to keep the water-level at
A the same as before, the cut AB must be made smaller than
the original channel, but the velocity in it will be greater, and
there will therefore be a still greater tendency to scour. If the
abandoned loop is left open the velocity in it will be greatly
reduced, owing to the lower water-level at A, and at B will be
further reduced by heading-up. It generally silts up.
To increase the discharge of a channel ABC (Fig. 136, p. 266),
supposed to be of shallow section, without enlarging it through-
out, the plan involving least work is to alter the bed to DB. As
D recedes from A the discharge increases, but so does the tendency
to silt. (Cf. chap. vi. art. 2.)
9. Effect of a Weir or Raised Bed. — The tendency to silting,
common to all cases of heading-up, may be somewhat enhanced in
the case of a rise in the bed or a weir extending across a channel,
because of the obstruction offered to rolling material. This how-
ever does not seem to be very great. The silt may form a long
slope against the weir, and material may be rolled up the slope.
Usually even this slope is not formed. Probably the eddies stir
up the silt, and it is carried over.
The deposit occurring upstream of a rise or a weir has caused it
to Ijo supposed that there is a layer of still water upstream of and
OPEN CHANNELS — VARIABLE FLOW 263
below the level of the crest. This idea is absolutely untenable.
The general velocity undoubtedly decreases as the rise or weir is
approached. This is due to the increasing section of the stream.
If the water below DE (Figs. 124 and 128) were still the section
would be decreasing. The same amount of heading-up might be
caused by obstructions of other forms, but it has been shown,
(art. 4) not only that the water upstream of them is moving, but
that upstream of the eddies not even the distribution of the veloci-
ties is affected. The same is no doubt true of a rise or weir. If in
a silt-bearing stream the water near the bed were still, there would
be a rapid deposit of silt as there is in a short hollow or recess.
But the contrary often happens. In some of the large canals in India
the bed upstream of -bridges has been scoured for miles, to a depth
of perhaps two feet below the masonry floors of the bridges which
are left standing up, and forming, in fact, submerged weirs. This
alone shows the preposterous nature of the still-water theory.
The idea might have been supposed to be exploded, but for a
somewhat recent case. In a paper on the Irrawaddy ^ it is stated
that, if the discharges for the water-levels A, 0, etc. (Fig. 133),
are plotted, the discharge seems to
become zero at E, which is level
with a sand-bar four miles down-
stream, although the depth EQ was _g__^ p
34 feet, and that ' this dead area of ^,^^^^^^^^
cross-section lying below the level ^^^^^^^^^
of the bar regulating the discharge, ^^^ ^33
exists on almost all rivers.' It is
natural that the discharge should become zero at E. As the water-
level falls the effect of the obstruction at F increases (art. 7), and
the surface-slope becomes flatter. If the water-level ever fell to E
the surface would be horizontal and the discharge zero. But the
reduction of the discharge to zero is due to the flattening of the
slope, and not to a portion of the section of the stream being
still. If it were still it could never have been scoured out, or
being in existence it would quickly silt up.
' Profile walls ' are sometimes built across a channel at intervals.
They are useful for showing the correct form of the cross-section,
but will not prevent scour, unless built extremely close together.
A single wall built at a point where the bed-slope becomes
steeper will not prevent scour. If scour does occur, walls or
weirs will of course stop it eventually.
' Minvies of Proceedings, Institution of Givil Engineers, vol. oxiii.
264
HYDRAULICS
In clearing the silt from a canal it is often convenient to make
the level of the cleared bed coincide with the level of a masonry
bridge floor, but it is not a fact that any deeper clearance is use-
less. The deeper bed gives an increased discharge for the same
water-level, and there is not necessarily a deposit of silt upstream
of the raised floor. Similarly, there is no particular harm in
omitting the clearance in any reach where, the depth of the
deposit being small, say half a foot, it is troublesome to clear it.
Section III. — Variable Flow in a Uniform Channel
(FormulcB and Analysis) ,
10. Formulse.^ — To find the length L between two points where
the depths are D^ and n,_ (Fig. 134)
let S' be the bed-slope. Then
h=n,-D^+LS'.
And from equation 17, p. 22,
yi
■(D,-D,_+LS'+K)
Pro. 134.
or r-L-C'BS'L=C'Ii(D^-D^+K).
Therefore . F^ITcW
where C, B, and F have values suited to the mean section between
the two points. The quantity A„ is nearly always small compared
to {D,-D,). In heading-up {D,-D,) and {F'-CBS') are nega-
tive, so that in equation 74 both numerator and denominator are
negative. In drawing-down the above quantities are positive.
To find the surface-slope S at any point, consider a point mid-
way between the two sections, and suppose them very near
together, so that the changes are very small. Let T\—l\=:r,
then F,'-FJ = (F+^y -(r-^^y = 21-1' and equation 17
^'^^ ^"^ .(75).
becomes h=-
G'B g
Let A be the sectional area and. 5 the surface- width at the mid-
way point. Let a be the difference in area in the length L.
vA Fa va
2 ~T~4'
aF
''A'
Then ()= F. t=.(F+l^(A- l^ = FA ^'
neglecting the very small last term, vA = I'a or
OPEN CHANNELS — VARIABLE FLOW 265
Therefore from equation 75, h=^-K!^. But a=B(D„-DA
and if A is the mean depth in the cross-section, A—Bd.
K'-5)-(<S-^>
Therefore S=^ = ^ . 9<^ . . . (76).
gd
The difference between the bed-slope and the surface-slope is
Q,_y_ \ gd) \CfR gd) CB /77X
1-— 1— —
gd gd
The fraction by which .— -, is multiplied in equation 76 is the
C It,
ratio of the surface-slope to what it would be in a uniform stream
with the same velocity and hydraulic radius. This fraction may
be written ^ where F' is the velocity in a uniform
gd
stream with the same values of and B, but with a slope equal
to the bed-slope. For ordinary depths and velocities the nume-
rator is not much less than unity. In cases of heading-up the
denominator is still nearer unity, but in drawing-down less so.
In a stream of shallow section B is nearly as d and J^is as --=-, so
that, neglecting the above fraction S is for moderate changes in
depth roughly as -^. In order that the slope obtained by
CI
observing the water-levels at the ends of a reach may agree with
the local slope at the centre of the reach, the sectional areas of the
stream at the two ends of the reach must not differ, in ordinary
cases, by more than 10 or 12 per cent.
Equation 76 establishes a direct connection between the depth
at any cross-section and the surface-slope at that section, but not
the connection between the depth or slope at any section and the
position of the section. To find this, the profile must be worked
266
HYDRAULICS
out in short reaches (restricted as above as to length) by equa-
tion 71, or by a method which will be given below.
To find the length of a tangent from any point K (Figs. 122
and 123, p. 255) to N, where it meets the line of natural water-
surface. Let D be the depth at K and D' the natural depth.
Let GN=x, GD=y. Then y = xS' and y+D'-D=xS.
Therefore D - D' ^x{S' - S)
and
»=
S'-S"
1-
■-(D-D')
gd
(78).
When the bed is level or slopes upward (Figs. 135 and 136)
Pia. 186.
Fio. 138.
S' in equations 74 and 76 is zero or negative. In the former case
j^_ CR{D,-D,+K)
8 =
1-
gd
(79)
(80).
11. Standing Wave. — If a stream has a high velocity relatively
to the depth of water in it V^ may be greater than gd. Let
heading-up occur in such a stream, so that J"' becomes less than
gd. Then the curve of heading up does not extend back till it
touches the natural water-surface, but ends abruptly at a point
A (Fig. 137). Ai this point V-—gd, the denominator in equation
76 is zero, and the slope therefore infinite, that is, the water-
surface is vertical, or a standing
wave occurs. In order that the
velocity may be sufficiently high,
relatively to the depth, to pro-
duce a standing wave, the slope
must be steep or the channel
smooth. It is not necessary that
there should be any variable
The flow in both the upstream and
Instances may be seen
Flo. 137.
flow cx'cept at the wave.
downstream reaches may be uniform.
OPEN CHANNELS — VARIABLE FLOW
267
where a steep wooden trough tails into a pond or downstream of
a sloping weir or contracted water-way. One occurs where the
Amazon suddenly changes its slope. The quantity
2^
equation 17 is greater than, and of opposite sign to the quantity,
-^^ . In order that V or C^RS may be greater than gd, S must
be greater than -^ assuming B and d to be equal. If C is 100, S
must be more than '0032.
At the foot of a rapid forming the left flank of the weir across
the river Ravi at the head of the Bari Doab Caniil the standing-
wave, when floods are passing, is 6 or 8 feet high, not counting
the masses of broken water on the crest of the wave. Logs
6 feet in diameter brought down by the flood disappear into the
wave.
The following statement shows some results observed by Bidone : —
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Dl
-D2
n
I'q
2?
-D2-D1
DifTerence
of Columns
5 and 6.
2s
Feet.
•149
•246
Feet.
•423
•739
4-59
6-28
1-62
2^09
•287
•545
Feet.
•274
•493
■013
■052
•137
•273
Column 7 shows (chap. ii. art. 1) the head lost. This is small and is nothing
like' 1—2' (chap. ii. art. 18), but it is much greater for the second case
than for the first.
Let AB (Fig. 138) be a stream, and let it be desired to lower
the water-level at E, say in order that floating logs or rafts may
clear a structure C, or in order to allow of a drainage outfall into
268-
HYDRAULICS
the stream. The object can be to some extent attained by head-
ing up the stream and introducing a rapid DE. It is conceivable
that some practical application of this principle might occur.
(Cf. case of constricted pipe, chap. v. art. 7.)
A standing wave is also called a jump. The condition necessary
for its existence is that upstream (Fig. 138a) (fj<—l- , and that
downstream d„ > —2- . To find the height of the wave. Let the
9
bed of the channel be horizontal and the width of the stream be
unity. In a short time t let the mass mmpq come to the position
m'n'p'q'. The change of momentum is the difference between that
of mrvn'm! and of pqqp'. The force causing the change is the
Fig. 138a.
difference between the pressures on mn and on qp.
impulse and change of momentum,
Equating the
wu.
d.
-d,^^]t-.
IP
(rf,lV-rfiTV)(
d,'-d,^^±{d,V,^-d,V-^).
But di Fi = d^ V^ and V^^ = ]\^'^ .
d^
Substituting this value of V^ in the above,
2
d,^-d,^ = -
-H^^i"')
{d,-d,){d, + d,) = 2{d,-d.£^-
Multiplying by ^_^^,
(d^+d„)d^ ^^]y-
Whence the quadratic
d,-^ + d,d,+'^f=2d,^ + ''f.
' 4 a 4
r,s
OPEN CHANNELS — VARIABLE FLOW 269
Therefore
"'-^^V-f+f
<i,-J?iz^l''^y^ . . . (80.)
which gives d^ in terms of d^ and V-^ .
The first term on the right in equation 80a is by far the greatest,
and (ig is more affected by change in Fj than by change in dy
The stream of high velocity necessary to cause a standing wave
can be produced not only by means of a steep or smooth channel,
but by means of a falling sheet (Fig. 74, p. 95) or by the issue of
the stream under a head (Fig. 63, p. 69). If the shoot there shown
be supposed to have ita water level raised, say by means of a weir,
to the proper level, a standing wave will occur. The discharge, Q,
is then independent of the level of the tail water. Further raising
of the tail water alters the conditions and Q depends on E-^ - 11^ as
usual. Experiments by Gibspn ^ on flow under a sluice gate show
Fig. 138b.
that the height of the standing wave was nearly as given by
equation 80a. Also that, if the tail water, instead of being raised,
is lowered there is no standing wave, the water rising gradually
(as at XY, Fig. 9, p. 14) a marked instance of the rises — frequently
referred to in the present work — which may occur in variable flow.
The surface is in this particular case convex upwards although the
depth is increasing.
In any standing wave a large amount of energy is absorbed.
There is much tumbling of the water and a general foamy condition.
There is, of course, loss of head. Otherwise the preceding proof
would not be needed and the increase in pressure head would be
equal to the decrease in velocity head. Immediately below the
jump the water is raised to a level higher than that of the water
downstream of it (Fig. 138b), but this 'superelevation' is not
dealt with in the above formula. It will be again mentioned below.
It has been seen that upstream of a standing wave V-^>gd or
Y "^ d
-!->—. That is, the velocity at mid-depth — and this is nearly the
^ Min, Froc, Inst. G.E., vol. cxcvii.
270 HYDRAULICS
mean velocity of the stream — is greater than would be attained if
the stream issued from an orifice undcir a head equal to the half
depth. Let y^ = gd^. Then Y^ is the critical velocity with
reference to a depth d^, that is the least velocity which can cause
a standing wave when the upstream depth is d^-
For given values of d-^^ and F^ a standing wave will be formed
only when d^ satisfies equation 80a. If, from any cause operating
in the downstream reach, d^ is increased and the bed of the upstream
channel is sloping, the jump shifts to a point further upstream.
It shifts downstream if the downstream water level is lowered.
If — as in the case shown in Fig. 138 — it cannot shift further
downstream, the jump is imperfect and there is great disturbance,
waves and broken water. The same thing may occur when there
is no well-defined channel downstream but merely a pond.
A jump, as above remarked, absorbs a very large amount of
energy, and the best method of preventing a large stream, issuing
say from a sluice, from doing damage is to construct a rapid and
cause the jump to occur, i The design of the rapid should be such
that the jump will occur at a suitable place and in a complete
form. The report just quoted describes new experiments made
with standing waves, d-^ ranging up to "22 foot, Fj up to 14:-9 feet
per second, and d^ up to 1"15 foot. The jump can occur even when
-i is as low as '025, but the position of the jump is then uncertain.
On rapids constructed in connection with irrigation works in
Burma the head on the crest may be 3 feet to 11 feet. The slope
of the rapid is generally about 1 in 15. It has been seen (chap, iv.,
art. 15) that the depth of water on the crest may possibly be about
the critical depth, d^. As the water flows down the slope its
velocity further increases and its depth decreases. At X (Fig. 80c,
p. 109) let the depth be d^. In this particular case the surface
is concave upwards although the depth is decreasing. At M the
depth is the natural depth and the flow has become uniform. N in
Kutter's ^ formula being known, the lengths of the curves can be
calculated as explained in article 12 and the profile of the whole
water surface obtained. Ifut experiments on large existing rapids
are first required in order to see what the exact conditions are. It
will then be easier to design other rapids on correct principles.
' State of Ohio. The Miami Conservanoy District, Teolinical Reports. Partiii.
Dayton, Ohio.
^ Probably about '020 fur a rapid pitohod with boulders.
OPEN CHANNELS — VARIABLE FLOW 271
The water level below the rapid being known, the position of the
standing wave can be found.
The slopes of the rapids are of boulders. The channel below
the rapid is protected by pitching (Fig. 138b), but not for any
great length. A rapid should be so designed as to reduce the
action on the pitching and channel. This is less the higher up
the jump occurs and the lower the velocity downstream of the jump.
If a rapid is roughened dj^ is increased, and V^ is reduced. Since
d^ depends more on F, than on d^, therefore d^ is reduced — that is,
the jump occurs higher up than before. On any given rapid an
increase in the discharge causes a rise in the downstream water-
level and the jump occurs higher up. The jump should be
complete for all except low discharges. Let the slope of a rapid
be produced so as to bring the crest further, upstream with a
reduced depth of water on it and let the length of the crest be
increased so that the discharge is as before. The downstream water
level is as before. The j ump occurs higher up. Fj, d^, and d,^ are
all reduced. Another plan is to splay out the side walls so as to
gradually increase the width of the rapid from the crest downwards.
If the slope of a rapid is steepened, F^ is increased and d^^ reduced ;
dj is increased. The jump occurs at a relatively greater distance
from the crest and actually nearer to the channel. The action is
more violent, and the rapid, though shorter, must be built more
strongly.
The superelevation at the jump is due to air and water being
intimately mixed so as to form a homogeneous mass not so heavy
as water alone. If the total depth at the wave is //„ and d^ is the
critical depth, then Hy] = Kdc. In the experiments referred to
above, K was found to be as follows : —
•i -5 -6 -7 -8
3-3 2-3 1-7 1-4 1-2
This information is useful for finding the height of the side
walls.
12. The Surface-curve. — In any given channel with a given
discharge there is only one curve of heading-up and one of
drawing-dovra, vi^hatever the cause of the variable flow may be.
If the cause operating at A (Figs. 122 and 123) be removed and
another cause introduced, say at K, making the water-level at A'
as before, the curve BK is the same as before. The water in the
'^i- 2
d.
•25
•3
■35
K = 12-6
8-1
5-7
4-3
272 HYDRAULICS
reach HK is only concerned with accommodating itself to the
water-level at K, and not with the question how that water-
level has been caused. If the surface-curve is once found, it will
not have to be found again for any lesser change of water-level,
but only a part of the same curve used. Theoretically the curve
extends to an infinite distance upstream, approaching indefinitely
near to the line BCt, which is an asymptote of the curve. Practi-
cally the curve extends to a limited distance beyond which no
change in the natural water-surface is perceptible. The less the
ratio of KD to -O'the greater is the relative length of the curve BK.
If the discharge of the channel is altered, the curve is entirely
changed, and no part of it is the same as any part of the original
curve. If the natural water-level is higher than before, a change
of the same amount as before will cause a smaller ratio of KD to
KF, and therefore a longer curve. The greater the relative area
of that part of the cross-section of a stream which lies over the
side-slopes of the channel, the more rapidly does the section
change with change of water-level, the more, therefore, does the
surface-slope at K differ from the natural slope, and the less the
length of the curve. The length of the curve is of course less
the steeper the bed-slope.
The curves for heading up are far more important than those for
drawing down. Heading up is frequently caused by weirs or
obstructions or by swollen tributaries or flood-water entering a
stream, and the effect at upstream points is often important.
Drawing down is far less frequent, and when it occurs is generally
of less consequence.
In all cases met with in long uniform channels the curve is
concave upwards when the depth is increasing and convex upwards
when it is decreasing. But when the depth is less than d^ the
rule is reversed, as stated in art. 11.
13. Method of finding Surface-curve. — To obviate the tedious
process of working out length by length, and obtain a direct
approximation to the surface-curve, one or two methods have been
used. An old rule, given by Neville for cases of heading-up, is
that the total length of the curve BK (Pig. 12-2, p. 256) is 1-5 to
1'9 times the length of the horizontal line KM. This is only an
approximation, or rather guess, of the very roughest kind, and it
gives no idea of the form of the curve, that is, of the depths at
intermediate points. For an imaginary case in which the bed-
width is infinite, the sides vertical, and the co-efficient C constant
for all depths, an equation to the curve can be found by integration
OPEN CHANNELS — VARIABLE PLOW
273
It is far too complicated for practical use, but certain tables have
been based on it. Such tables, owing to the wholly imaginary condi-
^'^|S uoc'os
274 HYDRAULICS
tions of the case, are of very limited use. For channels with vertical
sides they are not accurate, for others not even fairly accurate.
Fig. 139 shows four curves worked out length by length by
equation 74 (p. 264), for streams 5 feet deep with a slope of 1 in
4000, the co-efRcient C being about 60 when the depth is 5 feet.
For other depths the co-efficient is suitably increased. The curves
all tend to become straight lines as the depth increases. This is
owing to the minuteness of the surface-slope at great depths. The
fall in GF has a great relative difference to the fall in FA, but
both are so small that the divergence of the curve from a straight
line is sometimes imperceptible. The curves are drawn up to a
depth of 10 feet in one direction and 5 ■125 feet^in the other.
Below this depth the curve again tends to become straight. The
three uppermost curves are for channels of rectangular section.
' The uppermost curve represents the extreme limit possible, the bed
being assumed of width zero, or, what is the same thing, assumed
to be quite smooth, the sides being only taken into account in
calculating i?, which is therefore constant. In the second curve R
increases from 2-50 feet to 3'33 feet. The third curve is for
a channel of infinite width, but it is not the imaginary curve
mentioned above, because the co-efiicient C has been increased as
B increases, instead of being constant. As R increases from 5 to
10 feet R also increases from 5 to 10 feet. In channels with
sloping sides increase of depth is accompanied by a rapid increase
of section and of R and C. The profiles curve more rapidly, and
the points where the curves become straight are sooner reached.
The lowest curve is for a triangular section (bed-width zero), and
represents the extreme limit possible. For greater bed-widths the
effect of the side-slopes becomes less and vanishes when the bed-
width is infinite. The third curve, therefore, represents the other
limit in this case. The surface-slopes at A are, for the four curves,
isk' 24.481' 4»,W ^'^^ mm- *« last ^oii^S oiily Ath of the slope at B.
The total length of the curve — up to the point where D=\ •0252)'
—is 2-538, 2-057, 1-732, or 1-382 times the length of the horizontal
line AQ. The heading up at Q is -375, -313, -234, or -164 of the
heading up at A.
It will be seen directly that as long as the proportions of the
channel are maintained — even though its roughness or gradient
may alter — the curves, including the particular ratios just mentioned,
remain in most cases essentially the same. For a large number
of cases it will suffice merely to take the depths by scale from one
' That is, to 1 -025 D'.
OPEN CHANNELS — VARIABLE FLOW
275
of the curves of Fig. 139 — or any part of it — or any intermediate
curve that may be estimated to suit the case. The vertical and
horizontal scales of the diagram can be altered without altering
the actual diagram.
It will be useful to consider these curves further. Cross
sections of the streams corresponding to the four curves of Fig. 139
are shown in Fig. 139a. The increase of C^R as D increases from
Section
Ratio
(Table LI.).
Reference
to
Fig. 139.
1st curve
Increase
inC^iJ
(per cent.).
2nd curve
50
^^
83
Infinity 3rd curve 176
■75
zero 4tli curve 176
Wiiiiiiiiiiiiii^
Fig. 139a.
D' to 2D' is also shown. In equation 74 let h^, which is generally
very small, be neglected. Then
^-^§^^""'' ■•<«»■)
Consider channels with vertical sides. As D increases from D' to
21)', V^ is reduced by 75 per cent. The numerator and the first
term in the denominator of the above fraction both increase at the
same rate. When D only slightly exceeds D', V- is only slightly
less than C^RS', and the denominator of the fraction is far
less than C'^RS'. When D is about 2D', V^ is small and the
276 HYDRAULICS
denominator greatly increased. T^hxis L, for a given value of
Z>2 - i)j, decreases as Z>' increases and tends to become constant.
The greater the bed width of the channel the greater the rate of
increase of C^B, the less the relative value of V'^ and the less the
value of L. This is especially the case when D is great. Con-
sidering, say, the second and third curves, the lower one has every-
where the lesser value of L, but the difference is greatest when I)
is greatest. The two curves are essentially diiFerent.
The equation obtained by integration and referred to above is : —
-v-'--a-?)iC')-K§)}
The function <j> — called the backwater function — is complicated,^
B'
but values of it are given in tables for various values of — . For
the usual flat slopes — is only a small fraction of -^, so that L
depends very little on C. It depends almost entirely on (D^ — -Oj)
and obviously cannot be correct for the various ratios of width
to depth. The value of L obtained by using it may be wrong,
even though the value taken for C^ may be selected so as to suit
the stream in question.
For a channel of triangular section Ji increases at the same
rate as in the case represented by the third curve, being doubled
when D' is doubled, but A is then quadrupled and V- is reduced by
about 94 per cent. The reduction of L for great depths is more
marked. In using the backwater function tables for channels with
sloping sides D is taken as the sectional area divided by the
surface width, but even in this case the results are liable to be
quite wrong.
In oases where scaling from the diagram is not sufficiently
precise the procedure may be as follows. From equation 74 (p. 264),
1 F^" S'
I"" CRiD.-D.+K)' D,-D,+K ' ' ' ^^^^'
Let a:' = ' ~ — -, then x' is the length in which the bed-level changes
by (Di — Di) feet, and L is the length in which the depth changes
by (Di — D.i) feet. If the ratio ' is known L can be easily found.
This ratio, for ivxch of the uliove turves (except the uppermost,
which is not needed) and for some intermediate cases, is given
approximately in table li. for a ranfi;o of depth extending up to
OPEN CHANNELS — VARIABLE FLOW 277
D'
22)', the value of (Z), — D^) being usually y^, which gives reaches
sufficiently short to enable equation 74 or 81 to apply without any
considerable error. The approximate ratios -j are easily found by
disregarding K. Then, putting G'R8'= F", from equation 81,
This quantity, since D.>D^, is negative, and in table li. the
quantity 1 — =^2 is shown instead.
Now the ratios -=- in table li. apply, not only to the cases from
which they were deduced, but to a very large proportion of
other cases. Let the size, roughness, or bed-slope of the stream
alter in any manner, the proportions of the stream being main-
tained, and the proportionate change in with change of li being
also maintained, and let 'j, " be as before, then ™ and j
are as before. Thus the ratios in table li. can be used, with
suitable interpolations, for any channel whose section is rectangular
or trapezoidal. For a curvilinear or irregular section the section
most resembling it can be adopted.'^
Still greater exactness can be obtained as follows : —
Denoting by C-^ the value of G for the natural depth D', and C^ the value
for the headed-up depth 2D', column 14 of table li. shows the ratios — ^
or M, which actually occurred in the cases worked out. These ratios are fair
averages, being such as occur with streams 5 feet to 10 feet deep with N
about '0275, but for other cases the ratio may be different. For a very
smooth deep stream it will be less, and for a rough shallow stream more.
For-values of R (in the reach of natural flow) ranging from 2 feet to 8 feet,
and N ranging from '017 to -030, the value of M (Kutter and Bazin) may
possibly vary as shown in columns 15 and 16. For any given stream it will
be difficult to say what the value is, and the extreme values shown are not
likely to occur. Suppose that, for the second case shown in table li., it is
believed that M' is 116. Then '^'=1^ = 1-055 and :^',^=1-11 nearly.
M 1-10 M^ •'
Corrections can be applied as follows : —
Column of table li. : 3, 4, 6, . . 11, 12, 13
(-In C^iJ or F'2 ( -I- ) say, 4,1, 2, . . . 9, 10, 11 percent.
Correction.Jl'^r^''"*-)^^^' 1,1, 2,... 8, 9. 10pcrcent«
[in J (-f-) say, 4^,4, 4, . . . 2i, 2, 2 per cent.
The correction to be applied to ^ is -H or - according as -j-^ is > 1 '0 or
<10.
■^ For recent tests of tables li. and lii, see Notes at end of chapter.
= Since 100-rl'll = 90 nearly.
278
HYDRAULICS
For trapezoidal channels table li. gives the ratio —^, but the channels
At
oonoerned had side-slopes of 4 to 3. For other side-slopes the increase of if,
even with the same value of -r~, may differ somewhat, but the difference is
likely to be considerable only for a deep narrow channel. In any case a
p 2 J?
correction can be made, as above, by considering the change in 7,^-5^
instead of in -i. The actual values of R^ and -Rj were as follows •■ —
Oi
3
3-64
6 25
•75
2-69
4-78
Section ratio = Infinity
R^ = 5-0
ifj = 10-0
Regarding the hitherto neglected quantity h„ the following table shows
such values of it as have been worked out for the above cases. Except with
2-0
1-72
1-78
0-0
2-0
4-0
2
Values of A„.
Section
Ratio
(see
table li.).
2
4
lufinity
3
•75
0-0
Ve-
locity
where
depth
is
5 feet.
60
1-73
2-12
1^81
1-56
2'68
Depths of Water.
5-125
5-25
6.6
6
6-5
7
7-5
8
8-5
9
9-5
to
to
to
to
to
to
to
to
to
to
to
6-25
6-5
6
6-6
7
7-5
8
8-5
9
9-5
10
Values of Di-fla.
125 -25 -6 -6
025
003
•046
■006
:oi3
•074
•006
■009
•008
•007
•023
•058
■005
•007
•006
•015
■046
018
•036
■0046
•007
•030
•0025
•005
■026
■0023
■0015
■004
■021
•018
■0014
■015
•0013
•002
•001 { ...
... -0010
high velocities /i„i3 small compared to (D^- D^). For a smaller channel
(Z»i-Z>2) will be less, but probably T'and 7t„ will also be less. By inter-
polating and noting that K is as I'^ the values of h, for any case can
be approximately obtained and y corrected by multiplying it by
I) 2 n +}i ' '^^^"^t since D2> Di, is greater than unity, so that the correc-
tion increases
7/
Ordinarily the corrections have little effect, because D changes
less rapidly than j. Suppose the ratio -= used is wrong by 4 per
OPEN CHANNELS — VARIABLE FLOW
279
cent., then instead of giving the point where D is, say, 1 -30, it gives
the point where D is 1-28 or 1-32.
The profile can be easily extended with accuracy to a point
where the depth is greater than ID' by simply calculating the
surface-slopes at the two ends of the extension and drawing two
straight lines or even one.
Table Hi. shows some co-efficients j for cases of drawing-down
extending to half the natural depth. As with the curves of heading-
up the greatest change of slope and the shortest curve occurs
with a channel of triangular section. Fig. 140 shows one of the
tC—
Fia. 140.
curves. The channels are the same as before, but the natural depth
D' is now 10 feet, so that column 1 is not as before, and D^—D.^
. D'
20
D'
Ci now refers to the depth D' and O^ to the depth _. The correction
to be applied to '— for change in M is, as before, + or
according as
M
is > 1 '0 or < 1 '0, but it is greater than before in relative amount. The
values of —J for the trapezoidal channels are the same as the values of
-S3
— ? given above. The correction for hv is the same as before, and, as before,
' x"
has the effect of increasing — .
Ij
Where D is not much less than D' the surface-curve is very
similar to that of heading-up, with similar proportionate depths ;
but as D decreases the resemblance ceases, and the curvature
increases rapidly, a tangent to the curve tending to eventually
become vertical instead of horizontal as in heading-up.
The ratios in tables IL and lii. have been arranged in the forjn
280 HYDRAULICS
given so as to admit of corrections being applied, or at least to
show how the corrections affect them. Otherwise it would be
L X
more convenient to show — instead of -^. It is, however, easy
X L
to convert the figures. If they are converted and L is great it can
be found once for all by adding up the various values of — and
multiplying by x.
14. Calculations of Discharges and Water-levels. — When the
flow in a reach is not variable throughout, the discharge can be
found from the depth — or vice versd — in its upper portion, and thus
Fis known. Then, the depth at the lower end, or at any point
in the variable length, being also known, the surface-curve can be
found by the method of the preceding article.
When the flow is variable throughout a reach, such as AK
(Figs. 122 and 123, p. 255), supposing a breach in uniformity
to occur at K, an approximate discharge can be found by the
formula for uniform flow, the slope being KA and the depth being
greatetf or less than the mean of the two depths at K and A,
according as draw or heading-up exists. The reach can then be
divided into a few lengths, or left undivided (according as the
relative difference in the two depths at K and A is great or small),
and a nearer approximation made by using equation 74. If the
depths at K and A are very different the channel can be assumed
to extend up to B and table li. or Hi. used. In any case the
correct discharge is obtained when, the water-level at one end being
assumed, that at the other end comes out correct.
Whether or not the flow is variable throughout the reach, if
the discharge is so great as to affect the original water-level at the
head of the reach, allowance must be made for this in assuming the
water-level at B or K.
A case occurred ^ in which a cut, BA, with a level bed (Fig. 135,
p. 266) connected two rivers. It was desired to ascertain how
much water would flow along the cut. The writer of the article
worked out the discharge from first principles by the aid of the
calculus, the working occupying several pages. This case, as well
as that shown in Fig. 136, can be dealt with as above, except that,
D' being infinite, tables li. and lii. cannot be used, and that for the
level bed equation 79 (which is simpler) is to be used instead of 74.
To find approximately the depth ^jY(Fig. 135) for which the
' Minnies of Proceedings, JnntUiition of Civil Engineers, vol. liii.
OPEN CHANNELS — VARIABLE FLOW 281
discharge will be a maximum, BM heing given, let BM—D and
NA=i/. The section CQ is nearly as ^l^, Jli as J^^,
and ^S as / — ^. Then assuming C constant, Q is nearly as
^ = constantx{{D'-y)i-y{D+y){D'-f)i};
—constsmtxiB^—l/^—By—y').
When the expression in brackets is zero y-\ — -t-~ ^'~^^-
The discharge is a maximum when y= — and a minimum when
y=D. The discharge, however, varies little for a considerable
variation in y. In the case just referred to, when B was 8 feet,
the discharges found were, being constant,
2/= 1ft. 2 ft. 3 ft. 4 ft. 5 ft. 6 ft.
Q= 249 253 255 259 240 229.
Similar interesting problems occur on Inundation Canals, though,
owing to the temporary nature of the conditions, approximate
solutions are sufficient. When the head-reach of a canal is silted
and the time is approaching when the canal, owing to the falling
of the river, will go dry, a reserve head-channel is often opened.
Sometimes the first one is also left open. Whether it should be
left open or not depends on what extra supply it will give (when
the water-level at the junction is raised by the opening of the
reserve head) and on whether the slope in it will be so flat as to
cause it to silt excessively. If only one head is to be open it is
sometimes better to keep the reserve head closed, as the slope
along it may be flat owing to the conditions in the shifting river.
On the Choa branch of the Sirhind Canal the water, four miles
from the head, was headed up in order to work a mill, and the
variable flow extended up to the head, thus vitiating the discharge
table which depended on the reading of the head-gauge. The use
of the table was abandoned, but it would be possible to correct it
on the above principles, a gauge above the mill being also read.
The case of a silted canal head (art. 8) is difierent because the bed
is constantly changing.
Section IV. — Variable Flow in General
15. Flow in a Variable Channel. — Sections ii. and iii. of this
chapter treat of uniform channels, but though the propositions
282
HYDRAULICS
are more easily stated and proved for uniform channels, they
apply with certain modifications, which will readily suggest them-
selves, to variable channels. In uniform channels ' natural flow '
and 'uniform flow' both have the same meaning. In a variable
channel, if the water surfaces corresponding to various discharges
are termed the natural surfaces, and if ' natural flow ' is substituted
for 'uniform flow,' nearly the whole of section ii. applies. For
instance, if a weir is made, or a branch opened, the flow down-
stream of the alteration is still natural. The causes of variable
flow described in article 5 may be causes of heading-up or drawing-
down, or they may counteract each other, leaving the flow
natural. .
Generally a variable channel is in actual flow, so that the water-
level, for at least one discharge, can be observed. One problem
is to find the change of water-level which will be produced by a
change in the channel. The only way of finding the surface
profile exactly is to divide the channel into short lengths, in
each of which the section is either uiiiform or is varying in one
direction, and to use equation 74 (p. 264), which then applies. If
the channel is so variable as to consist of a number of pools and
rapids, the effect of a change of level at any point will often
extend back only to the next rapid.
The surface-slope at any point is always given by equation 76
(p. 265), that is, roughly, by the equation F= C JBS, where S is
the surface-slope. At any selected point let B be the width of
the water-surface and d the mean depth. Then roughly
Q=AV=.BdO JBS, or S=^^^
Since Bd=A, therefore S changes in the opposite manner to A.
Fig. 141 represents a case in which B is constant. Here d, R,
FlQ. 141,
and C all change in the same manner, and the changes in S are
very great. The vertical lines mark the points where it is a
niiiximum or minimum. The convex and concave surface-curves
OPEN CHANNELS — VARIABLE FLOW 283
touch one another at these points. The changes in S follow those
in the bed, but are less pronounced. If, instead of ^, B is
supposed to vary, the profile is similar, but the changes in S less
pronounced. If Fig. 141 is supposed to be a plan of such a
channel, instead of a longitudinal section, the surface will still
be like AF. If the changes of width and depth both occur
together, and are of the same kind, the changes in S are greater.
If from any cause heading-up or drawing-down occurs at F the
surface will undulate somewhat as before, approaching the natural
surface towards A. The greater the depth of water in a channel
the less the effect of inequalities in the bed. A stream which, at
high water, has a fairly uniform surface-slope, may at low water
form a succession of pools and rapids.
It has been stated that in a channel of varying width the
discharge depends only on the least width, and that in clearing
silt all clearance beyond the minimum width is useless. These
statements are quite incorrect.
A stream may be so irregular in plan and section that the
direction of the current is not parallel to what may seem to be the
axis of the channel and the water-surface far from level across.
The irregularities, if examined, will be found to be developments
of those discussed under curves, obstructions, etc. Very often
the excessive irregularity occurs only at low water.
16. Uniform and Variable Flow. — Whether variable flow takes
place in a uniform or in a variable channel there are many degrees
of variability. When the variability is very slight all the results
found for uniform flow obviously apply, and the same is true,
except as regards formulte and exact calculations, when the
variability is great. It will be clear, on consideration, that the
discussions of chapter vi. all apply, even if two successive sections
are not quite equal or similar.
In a variable stream a short length I can generally be found in
which the flow is uniform. If observations are made in such a
length for the purpose of finding C, the formula for uniform flow
applies if S is the local slope. If the fall in I is very small the
slope observations are often extended outside it. This was done
in some of the Roorkee experiments on earthen channels, where
the stream, though of uniform width, varied much in depth. The
results seemed to disagree with Kutter's co-efficients, but when
allowance was made for the variable flow they agreed quite well.
No doubt similar error has occurred in many experiments. The
proper method in such a case would be to observe V and B over
Xhe same length as that for which S is observed.
284 HYDRAULICS
The surface-slopes at opposite banks of a stream are not gener-
ally equal unless it is quite uniform and straight.
17. Rivers. — A river, especially at low water, may be a series
of separate streams with numerous junctions and bifurcations.
The water-level in a side-channel CAE (Fig. 142) may afford only
a very poor indication of the general water-level
in the river. Suppose that with a good supply the
water-level at A is the same as that at B. If there
is silt in the channel CA—ih.e silt being deepest at
C — a moderate decrease of the river discharge may
cause a great decrease in the discharge of CA, or
even a total cessation of discharge. This causes
great difficulties in the matter of gauge-readings in
some Indian rivers. Suppose a gauge to have been
originally at B. If erosion of the bank sets in the
Fig. 142. gauge has to be moved, and sometimes it is difficult
to find another place (free from practical difficulties in the matter
of reading the gauge and despatch of readings), except at such
a place as ^ in a side channel. In floods, especially when the
sandbanks between the channels are submerged, there is a general
tendency for the water-surface to become level across, but it by no
means follows that it becomes so. When the deep stream is at
one side of the river channel the flood-level is nearly always
higher on that side than at the opposite side.
Since a small cross-section tends to cause scour and a large
one silting, it follows that every stream tends to become uniform
in section. The remarks made in articles 1, 2, and 8 also show
that it tends to destroy obstructions, to assume a constant slope,
and to become curved in such a way that its velocity will suit the
soil through which it flows. If a river always discharged a con-
stant volume its regimen would probably be permanent. It is the
fluctuations in the discharge that cause disturbance.
Notes to Chapter VII
Momentum (arts. 1, 2, 3). — The effect of the momentum of
flowing water is apt to be exaggerated. When a stream enters
a tank or lake its current is quickly destroyed. When a large
river enters tho sea its effect on the colour or saltness of the water
may be perceptible for a great distance, but this is because the sea
level at the mouth of the river becomes very slightly raised so that
currents are caused. These extend not only straight out to sea,
but to right and left.
OPEN CHANNELS — VARIABLE FLOW 285
In the case of a sharp bend in a large river statements are
sometimes made to the eifeot that the ' full force of the stream ' has
to be contended with. It seems to be implied that the momentum
of the great mass of water is the danger. The scour along the
concave bank is due to velocity, not to momentum. Sometimes it
is implied that there is a danger of the river taking a straight
course. This again depends on scour and velocity and is a rare
occurrence, except as regards changes occurring within the sandy
channel of a broad river.
JEqvution for Variable Mow (chap. ii. art. 10, and chap. iv.
art. 15). — In a cross-section of a stream the mean of the squares of
all the velocities exceeds the square of the mean. In the case of
the numbers 2, 3, 4, the one quantity is 967 and the other 9'0.
The proper percentage to be added to -L- — ?- can only be decided
by observation at the place, but can probably in the case of a con-
tracted channel ^ be taken at 1 1 or i^th.
Tests of Tables LI. and LII. — Curves of heading up and draw-
down for two concrete conduits, one rectangular and one circular,
have been worked out by Jameson. ^ For a rectangular section
7-08 feet wide, with D' = 2-875 feet (section ratio 2-46), the lengths
in which D increased from 4 feet to 4'5 feet and from 4"5 feet to
5 feet were 2370 feet and 2087 feet respectively. The figures
arrived at by using table li. are 2409 feet and 2157 feet. The
diiference is no doubt due chiefly to the effect of A„, which in the
conduit was quite appreciable.
The section of the second channel is shown in Fig. 139a, the
diameter being 7'3 feet, and the natural depth 3'4 feet, with a
heading up of 2-08 feet. The curve of heading up is practically
parallel to that for the rectangular channel. This was to be
expected, since the sides are nearly vertical and the relative increase
in sectional area and hydraulic radius nearly as before. In using
table li. for such a channel the section would be assumed to be as
shown by the broken lines.
In the case of the conduit of rectangular section above-mentioned
the lengths in which D — in a case of draw-down — decreased from
2-5 feet to 2 feet and from 2 feet to 1-5 feet were 1474 feet and
345 feet respectively. By table lii. the lengths are 1735 feet and
513 feet. The difference is again due to the effect of h^, this
^ Calculation of Flow in Open Channels. Houk. See chap. iv. art. 15.
^ Paper read at Inst, of Water Engineers, 5th December 1919.
286
HYDRAULICS
quantity — considering in each case the whole length, and not
dividing it up — amounting to -089 and -191, while {D^ — D^ is in
each case -SO. When D was 1'5 feet V was 5-3 feet per second.
With such high ratios of F to i> the correction for h^ is considerable.
Figure
FOB
i Example 1.
'a
10
1*764^ —ZOOO- - *j'^^^5^
I*. 5662' ->|
FlGFKE
B yoR
Example 2.
Examples
Example 1. — In the channel considered in example 3 of chapter
vi. a heading-up of r25 ft. is caused by a weir. What heading-
up is caused 2000 feet upstream of the weir ?
Also J,,
80x4-75 = 380
x'
Table xlv. shows ./ = 402-6 sq. ft,
A
sq. ft. .-. ./s='22-6 sq. ft. and -,'=17 nearly, so that -y lies
between the values for the first and second cases in the second
part of table li., and somewhat nearer to the first than the second.
OPEN CHANNELS — VARIABLE FLOW 287
Since S' = ^ and D,-D, = ^=-i1b ft..-. a;'=^'~^^-
5000 ""^ ^i-^.-jQ=*/u iu. .. ., = ^jr—
■475x5000 = 2375 ft.
The headed-up depth at the weir is 6 ft. = 4-75x 1'264. From
table li. -j- is about -550 when D^ is 1-2Z*' and Z>, is 1-3Z>'.
a;' 2375
Therefore i=:ggQ = ^ggQ- = 43 18 ft. The distance of the weir
downstream from the point where the depth is 1-20Z)' is
1-264- 1-200
l-30- T-20'^ ® P°^"*^ ^^°^ ^*'- "Pstream of
the weir is thus 764 ft. from the above point, and the change of
764
depth in this length is (l-30-l-20)X>'x ro,g = -018Z)', s6 that the
heading-up is (1-218- 1-00)J)', or •218x-475 ft., or 1-04 ft.
Corrections if applied to this case might alter the result by -01 ft.
Example 2. — From the stream considered in the first trial in
example 2 of chapter vi. a branch is taken off and discharges
120 0. ft. per second. What lowering of the water-level is caused
1500 ft. upstream of the branch ?
Table xlv. shows ^ = 356-3. Also ^,,=40x7-5 = 300 sq. ft.
A,
X
.-. ^s=56-3 sq. ft. and -^-=5-32, so that y lies between the
values in the first two lines of the second part of table lii. The
discharge below the bifurcation is 967 c. ft., and this is given by
a depth of 7 ft., so that the lowering is -5 ft.
Since S"=^^ and D^-D^=^=-d7b ft. - -" ^'~^^
5000 ' •"''"20" "'" •^'" - - ■- - s" -
•375x5000=1875 ft. The drawn-down depth at the bifurcation
is 7 ft.=7-5x-9S ft. From table lii. -^ is about -33, when Z*!
x" 1875
is -95Z)' and X>, is -90X»'. Therefore i = :gg=-7gg- =5682 ft.
The distance of the bifurcation downstream from the point where
-95 — -93
the depth is -95i)' is .Qg_.QQ x5682 = 1894 ft. The point 1500
ft. upstream of the branch is thus 394 ft. from the above point, and
394
the change of depth in this length is (-95 — -90)Z>'x K/;Qr, =
•003472)', so that the drawing-down is X''-(-95 — •0035)Z»' or
■0535x7-5= -401 ft.
288
HYDRAULICS
Table LI. — Ratios for calculating Profile of Surface
WHEN headed up. (Art. 13.)
(1)
(2)
(3)
(4)
(6)
(6)
(T) (8)
(9) (10) (11) (12)
(13)
(14)
(15)
(16)
Depth Ratios Upper figures show ^, lower figures -p.
Values of M or &
Section
Ratio.
Ratios.
r025
1^06
I^IO 1-20
1^30
1^40
1^50
1^60
1^70
1^80
1-90
Actual
Extreme
Values.
to
to
to
to
to
to
to
to
to
1-05
I^IO
[■20 1-30
1-40
l-.'iO
1-60
1-70
I ■SO
1^90
2-0
oc-
curred.
Maxi-
mum.
Mini,
mum.
Rectangular Sections. Ratio of Width to Depth as in column 1.
c
V^^ ya
■903
•820 ^682
•648
•448
•371
•313
•267
•229
•199
•1751
2 [
,
\
107
1^12 10-2
\
^orl
-(1724-F'2)
■097
•180
■318
•452
•552
•629
■687
■733
■771
■201
•801
•171
•825J
(
p^.^y»2
•892
•804
•659
•516
•416
•336
■280
•234
•149 -v
4 [
\
110
1^16 103
\
^orl
_(72^F'2)
•108
•196
•341
•484
■584
■664
■720
■766
•799
•829
•851 j
In- r
finity'\^
F2-^F'2 -880
•m
•614
•458
■351
•274
•218
■175
•148
•118
•099^
^-
-Cn^v-)
•120
■223
•386
•542
•649
•726
•782
•825
•857
•882 •901/
117
1^28 'r05
i
A h
Trapezoidal Sections. Ratio -r—
As
area of section over bed
as in column 1.
~ area over side-slopes '
In-
finity
(The figures are the same as for the preceding case.)
\-Xi
1^28
1^05
(
^2^ ^^'2
•864
■760
■coo
•440
•334
■257
•198, ■164
i !
■122, ■098, ■OSO-l
a 1
>
V13
V21
V04
\
±orl
_(172-^.F'2)
•136
•240
•780
•400
•643
■560
•870
■666
■269
■743
■8021 •846
•878 -902
•920J
(
724. F'2
•847
■194
•146 109
■082 ^064
•052 -v
■75 \
i;-
1 (
i
1^13
\-'>A
1-04
I
(
_(T^^F'2)
•16a
•810
■270
■072
■457
■4'20
■630
■■J92
•731
■19-J
■S06 ■851-891
■130 ■090 OO-I
•918 •pse
■948/
^2^1-.
■Oio' •034
■026 -v
O'O \
1 1
}
MS
V2R
V05
^orl
- ( T'2-=- F'2)
■18fi
•328
•!>H0
•708
■808
■870' ■910-936
•954 •gee
■974 j
OPEN CHANNELS — VARIABLE FLOW
289
Table LII. — Ratios for calculating Profile of Surface
WHEN drawn down.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
Suction
Ratio.
Ratios.
Depth Batios.
Upper figures show _^, lower figures — ^,
Values of ilf or ^.
Ci
•95
to
■90
■90
to
■85
•S5
to
■80
■SO
to
■75
■75
to
•70
■70
to
■66
■65
to
■60
■60
to
■56
■66
to
■50
Actual
which
oc-
curred.
Extreme
Values.
Maxi-
mum.
Mini-
mum.
Rectangular Sections. Ratio of Width to Depth aa in column 1.
>{
^orF2-=-r2-l
1-21
■21
•39
vm.
•62
1
1^90 2^25
■90 1^25
1
2^72
1^72
3-33
2^33
4^14
3^14
5 ■SI
4^31
|-935
•89
•98
^{
?!orFii.^F'2-l
1-24
•24
1-31
•31
1^46
•45
1-69
•59
1-69
■69
1^94
•94
2^02 2^42
1-021 '42
1
2-42 3-04
1
1-42 2-04
1
3 '00
2^00
3^89
2^89
3^72
2^72
6-07
4^07
4^76
3^76
6^24
5^24
[•909
•86
•97
In- r
finity|
72^ 7'2
^'or72^F'2-l
6 ■so
5^80
9-36
8-35
[-85
J
•78
•95
Trapezoidal Sections. Ratio —7
_ area of section over be
-, as in colui
nn 1.
area over side-slopes
In-
finity
{The figures are the same as for the preceding case. )
■85
•78
■95
■375/
1-34
•34
1^07
•67
2-10
I^IO
2-30
1^30
2-84
1^84
2-70
1^70
3 ■08
2^08
3-99
2^99
3^60
2^50
4^12
3^12
5-77
4-77
4^56
3 •55
6 ■69
4^69
8-49
7^49
6^00
5^00
7 ■84
6^84
12^94
11-94
8^18
7*18
11 ■so
10^50
1S^62
17 •62
11-35
10^35
17^17
16^17
31^32
30-32
[-88
•83
-96
V^-i-V"^
Li
■36
1-73
•73
[-88
•83
■96
0-0 1
i'orF2-^F'2-l
1^52
•S2
2^06
1^06
}«
•78
•95
CHAPTER VIII
HYDEAULIC OBSERVATIONS
[For general remarks on Hydraulic Observations, see chap. ii. art. 25]
Section I. — General
1. Velocities. — When the velocity is observed at one or more
points in the cross-section of a stream, the process is termed ' Point
Measurement.' When the mean velocity on a line in the plane of
the cross-section is found directly, it is known as an ' Integrated
Measurement.' Velocity-measuring instruments are of two classes,
namely, 'Floats' and 'Fixed Instruments.' Fixed Instruments
give the velocities in one cross-section of a stream. Floats give
the average velocity in the ' run ' or length over which they are
timed, and not that at one cross-section. Floats are used only in
open streams, but fixed instruments sometimes in pipes.
With most instruments time observations are necessary. The
best instrument for this is a chronometer beating half-seconds,
similar to those used at sea, or a stop-watch which can be read to
quarter-seconds. The next best is a common pendulum swinging
in half-seconds, and after that an ordinary watch. The error in
timing with a chronometer is not likely to exceed half a second,
with an ordinary watch it may be one or even two seconds. Some
stop-watches and watches not only do not keep proper time, but
are not regular in their speed. If any such defect is suspected
the instrument should be tested. The time over which an obser-
vation extends should be such that any error in timing will be
relatively small. In order to eliminate the ' personal equation ' of
the observer similar observations at the beginning and end of the
time should bo performed by the same individual, or if performed
by two they should frequently change places.
Floats include surfac^e-floats, sub-surface floats, and rod-floats.
The first two are used for point measurement, the last for inte-
grated measurements on vertical lines. A float travels with the
stream, and so interferes little with the natural motion of the
HYDRAULIC OBSERVATIONS 291
water. Its velocity is supposed to be the same as that of the
water which it displaces.
Fixed Instruments are divided into Current Meters and Pressure
Instruments. In the former the velocity of the stream is inferred
from that of a revolving screw, in the latter from indications
caused directly by the pressure of the water. ^ Velocities cannot
be obtained by Fixed Instruments until they have been 'Rated,'
that is, until it has been ascertained that certain indications of the
instrument correspond to certain velocities. Fixed instruments
interfere with the natural motion of the stream, but this need not
cause error. The disturbance is almost entirely downstream of an
obstruction (chap. ii. art. 21), and if those parts of the instrument
which are intended to receive the effect of the current are kept
well upstream, no difficulty arises, except perhaps in very small
streams. If a boat is used the bow can be kept pointing upstream
and the instrument upstream of the bow, a platform being made
to project over the bow. Even if the boat or instrument is so
large (which is not likely) relatively to the stream as to cause a
general heading-up, this will not prevent a correct measurement of
the discharge, though it may affect the surface-slope. In order that
disturbance may not be caused by moorings the boat should (unless
it is a steam-launch which can maintain its position) be held by
shore-lines. If attached by its bow to a pulley running on a trans-
verse rope, it can quickly be brought, by using the rudder, to any
required point. Another transverse rope serves to keep the boat
steady and, if divided by marks, shows its position. In a wide
stream containing shallows the ropes may rest on trestles placed
at the shallows. Where moorings must be used it is best to moor
two boats side by side, as far apart as practicable, and to work
from a platform between them, keeping the instrument well
upstream.
The choice of an instrument for velocity measurement depends
on various considerations. Floats require a regular stream, but
fixed instruments can be used in any stream. In comparing the
Current-Meter, or Pitot's Tube with Floats, regard must be had
to the design and quality of the instruments available, and to the
manner in which they were rated. Sub-surface floats are unsuit-
^ Further information concerning Fixed Instruments is given in Sections
IV. and V. , but the varieties and details are very numerous and cannot all
be discussed. There are many papers on these instruments in the Minutes
of Proceedings of the Institution of Civil Engineers and Transactions of the
American Society of Civil Ungineers.
292 HYDRAULICS
able when the stream is rapid or when there are weeds growing
in it, fixed instruments unsuitable when the velocity is very low.
For surface velocities alone surface-floats are, in regular streams,
the best instruments unless there is considerable wind. For
integrated measurements the rod-float is as good as any instru-
ment, provided the bed is even enough to allow of a rod of the
proper length, or nearly the proper length, being used.
The above considerations refer to accuracy only. As regards
the time occupied and the number of observers required, fixed
instruments generally have the advantage. In a discharge
measurement of a large river current-meter integration measure-
ments can be made while the soundings across the channel are
being taken. On the other hand, the time occupied in rating the
fixed instruments, their initial cost, and their liability to damage
or loss; especially in out-of-the-way places, may be very important
factors. If a stream is too wide to be reached at all points without
a boat, has no suitable bridge, but is still narrow enough for the
floats to be thrown in from the sides, and if no soundings are
required, float observations may take less time than others.
2. Discharges. — The discharge of any small volume of water is
best found not by measuring the velocity, but by letting the water
pass into a tank and measuring the volume added in a given time.
In this method nothing, or next to nothing, is left to assumption.
Whenever leakage, absorption, or evaporation occur, allowance
must be made for them. For very small discharges the water can
be weighed. The methods adopted for high discharges are as
follows.
The discharge of an open stream is usually found by observing
the depths and mean velocities on a number of verticals. Let
ABG (Fig. 143) be the mean velocity curve, and ADEFC a curve
Pia. Its.
W
whose ordinates are found by multiplying the depth on each
vertical by the corresponding velocity. Then ADEFO is the dis-
HYDRAULIC OBSERVATIONS 293
charge curve, and its area is the discharge. If floats are used the
velocities obtained are the averages in the run, and the depths
must also be averages in the run. The more numerous the
verticals the more accurate the result. For ordinary work ten is
a fair number ; for very accurate work, twenty. In the segments
AB, FO, near the sides the verticals should be nearer together than
elsewhere, because the ordinates change rapidly. The equal
spacing of the verticals in each segment is not essential, but it
simplifies the calculation, as it is only necessary to add together
all the ordinates in a segment — deducting half the first and last —
and multiply the sum by the distance between the ordinates. The
discharges of all the segments added together gives that of the
stream. If the number of equal spaces in a segment is even
Simpson's rule can be used, but ordinarily the results of formulae
such as this differ very little from those of the simpler rule.
Sometimes the spacing in a segment cannot be equal. If there
is in the cross-section any marked angle, whether salient or
re-entering, a measurement should be made there. Sometimes
when floats are used in rivers the velocities must be observed
where the floats happen to run. In such cases the depths at
these exact points need not be measured, but may be inferred
from those observed at fixed intervals or found by plotting the
section.
If the mean velocity on a vertical is obtained by multiplying
the observed surface velocity by the co-efficient (3 (chap. vi. art. 9),
and if /3 is the same for all verticals, the discharge may be calcu-
lated as if the surface velocities were the means on verticals and
the whole discharge multiplied by /?.
Discharge observations in an open stream are greatly facilitated
by the construction of a 'Flume.' A short length of the channel
is constructed of masonry or timber. The sides may be sloping
but are preferably vertical. In the absence of silt deposit the
section of the stream is known from the water-level, and if rod-
floats are used they are all of one length. Flumes may, however,
prevent proper surface-slope observations (chap. vii. art. 5).
Discharges can be obtained with more or less exactness by the
observation of U or 11^ and the use of a or S (chap. vi. art. 10),
but a ilume may be unsuitable for this (chap. ii. art. 21) if there
is any abrupt change at its upstream end.
When the velocities in the whole cross-section of any open stream
cannot be observed, and even the approximate method just
mentioned is impracticable — as, for instance, in the case of a flood —
294 HTDKAULICS
the velocity is calculated from the surface slope and cross-section.
At the time of the flood, stakes should be driven in at the water
level, or other marks made. If this is not done flood marks on
trees or other objects should be observed in as great a number as
possible and discrepancies averaged. Flood discharges can also be
calculated from the water levels at bridge openings or contracted
portions of channel (chap. iv. art. 15).
Whenever discharges of open streams are observed it is highly
desirable to observe the surface slope and so ascertain C if for'
no other purpose than that of adding to existing information as
to co-efficients and values of N. But such observations cannot
usefully be made in any perfunctory manner. The greatest care is
required. In an earthen channel there is often the chance of the
sectional area varying within the slope length. The errors which
occurred in the Eoorkee Hydraulic Experiments have been men-
tioned (chap. vii. arts. 5 and 16). Preliminary longitudinal
soundings should if possible be taken over the whole slope length
or V should be observed at several places within the slope length.
If the channel is decidedly irregular, as in the case of many rivers,
several cross-sections should be taken within the slope length and
the mean value of V computed. Neglect of such precautions as
the above has led to remarkably erratic values of N being reported.
(See also art. 7).
The discharge, Q, of a small body of water can be ascertained by
"chemical gauging." A small and steady supply, q, of a soluble
material, say salt, is introduced into the stream. At a point further
downstream where thorough mixture has taken place samples of the
water are taken. A cubic foot, 62'4 lbs., is found to contain a
certain weight w of the chemical. The ratio of Q to q is the same
as that of 62'4 to w. Parker ^ mentions some practical difficulties
which may occur. He considers that, in order to obtain a steady
supply, a concentrated solution of the chemical must be made and
discharged through an orifice, but that, owing to impurities and
evaporation, the discharge will not be uniform unless the co-efficient
for the orifice is ascertained for each fresh batch of the chemical.
The discharge of a large pipe can be found by observing the
velocities by means of the Pitot tube (art. 14). The co-efficients
for orifices and w^irs in thin walls being well determined, these
arc frequently used as instruments for measuring the discharges
of small open channels or large pipes ; or weirs or orifices of kinds
other than thin-wall. The same is the case with the Venturi
» The Control of JVaier.
HYDRAULIC OBSERVATIONS 295
water meter (art. 16). In all the above cases the chief assumption
made is the value of the co-efficients — generally well known —
appertaining to the instruments or devices used.
When a discharge table has been prepared for any site or
aperttire the discharge can be found by simply observing the
water-level ^ or head or — in the case of a pipe — the hydraulic
gradient. The discharge of a pipe may be altered by incrustation
or vegetable growths, and that of a channel by changes occurring,
not only at the site but downstream of it. Frequent measurement
of the discharge may be necessary in order to correct the table. In
such cases the sectional areas and velocities should be tabulated so
that causes of error may be the more readily traced.
3. So'iiudings. — Soundings are generally taken to obtain a cross-
section of a stream, but longitudinal sections may be required in
order to find the most regular site, or in connection with float
observations. In water not more than about 15 feet deep
soundings are best taken with a rod, which may carry a flat shoe
to prevent its being driven into the bed. In greater depths a
weighted line is used.
Unless the velocity is very low it is best to observe soundings
from a boat drifting downstream. The current then exerts little
force on the rod or line, which can thus be kept vertical. It can
be held so as to clear the bed by a small amount, and lowered at
the proper moment. This plan is particularly suitable for obtain-
ing the mean cross-section in the run when floats are used. As
the boat drifts the bottom is frequently touched with the rod or
line, and the readings booked and averaged. Any local shallow
likely to interfere with the use of rod-floats is also thus detected.
When shore-lines can be used the boat can be worked and the
widths measured as described in article 1. In wide rivers lines
of flags or ' range-poles ' are used instead of ropes. An observer
on the boat or on shore can note the moment when the boat
crosses the line, and give a signal for the soundings to be taken.
To determine the distance of the boat from the bank an observer
in the boat reads an angle with a sextant, or an observer on shore
reads it with a theodolite, following the boat with his instru-
ment and keeping the cross wires on some part of it. When the
line is reached the motion of the instrument is stopped and the
angle read ofi'.
4. Miscellaneotis. — The diameters of pipes, while water was
flowing, were measured by Williams, Hubbell, and Fenkell by
1 This may be done by a self-recording gauge (art, 5).
296 HYDRAULICS
means of a rod with a hook inserted through a stuffing-box.
For obtaining the mean diameter in a length of pipe one method
is to fill it with water, which is afterwards measured or weighed.
If the joints are not closely filled in some error may be caused,
and Smith in some experiments filled each separate piece of pipe
before it was laid, and weighed the water it contained.
For ascertaining c, and c„ for orifices special arrangements are
required. The velocity of the jet is found by observing its range
on a horizontal plane. A ring or movable orifice of nearly the
size of the section of the jet may be placed so that the jet passes
through it, the flow stopped, and the necessary distances measured.
The actual velocity can then be found from equation 29 or 30
(p. 52), and, the actual head being measured, c„ is easily found.i
When observations of any kind are made a suitable form
should be prepared and filled in. It should have spaces set apart
for recording the date, time, gauge-reading, and (at least when
floats are used) the direction and force of the wind.
When extreme accuracy is required, as in the case of important
experiments, many precautions have to be taken. In small orifices
the edges have to be got up with very great accuracy. Excellent
work of this kind was done by Bilton (chap. iii. art. 8). With
a weir great care is necessary in observing the head. Nearly
all detailed accounts of hydraulic experiments, such as those
referred to in this work, contain instructive details as to methods
adopted.
Before undertaking any important experiments those concerned
should carefully study in every detail the instruments and methods
to be adopted and obtain preliminary practice with them.
On the question how far it is correct to disregard any co-efficient
or experimental result which seems to be abnormal, it is to be
noted that all observations made by any one person with equal care
and under similar conditions are entitled to equal weight. If
one experiment in a set gives a result greatly differing from the
rest, it is often rejected by the observer himself, the inference
being that there was a mistake, say in timing. This implies a
reduced degree of care in that observation. Whether the
difTerence is great enough to warrant rejection is a matter for
the judgment of tho observer. When it comes to an author
accepting or rejecting the result of an observation at which he was
not present, tho diflioulty is far greater because he does not know
all the facts. In some cases experiments have been rejected
' But see chap. iii. art. 9.
HYDKAULIO OBSEEVATIONS 297
without any reason being given, but apparently on the sole ground
that the results disagree with those of some other experiments.
Section II. — Water-levels and Pressure Heads
5. Gauges. — For observing the water-level of an open stream
the simplest kind of gauge is a vertical scale fixed in the stream
and graduated to tenths of a foot. It may be of enamelled iron,
screwed to a wooden post which is driven into the bed or spiked
to a masonry work. The zero may conveniently be at the bed-
level, so that the reading gives the depth of water. The actual
gauge may extend only down to low-water level. If a gauge is
exposed to the current it may be damaged by floating bodies, and
it is difficult to read it accurately, owing to the piling-up of the
water against the upstream face and the formation of a hollow
downstream. These irregularities can be greatly reduced by
sharpening the upstream and downstream faces of the post or the
upstream face only.i Greater accuracy can be obtained by placing
the gauge in a recess in the bank, but not where it is exposed to
the effects of irregularities in the channel (chap. vii. art. 2), and
by watching the fluctuations of the water-level, noting the highest
and lowest readings within a period of about half a minute, and
taking their mean ; but very great accuracy by direct reading of
a fixed gauge is difficult, because of the adhesion of the water to
the gauge, and the differences in level of the point observed and
the eye of the observer.
With floating gauges these difficulties are almost got rid of. The
graduated rod is attached at its lower end to a float which rises
and falls with the water-level. The rod travels vertically between
guides, and it is read by means of a fixed pointer on a level with
the eye of the observer. The float and rod should be of metal, so
that they may not alter in weight by absorbing moisture ; the
float perfectly water-tight and its top conical, so that it may not
1 Ward's Gauge, ■well known in India, consists of two vertical planks joined so
as to form an angle upstream. The gauge is placed between the planks on the
downstream side .
In a type of gauge used in tidal waters a pipe containing air extends down below
the water. As the tide rises the air is compressed. The recording apparatus is
actuated by a float resting on mercury in one leg of a U-tube, the other leg
being in communication with the pipe. The record can be made at a consider-
able distance away {Min. Proc. Inst. C.E., vol. cxcv.).
298 HYDRAULICS
form a resting-place for solid matter. The gauge should occasion-
ally be tested by comparison with a fixed gauge or bench-mark.
For a given weight of float and rod the smaller the horizontal section
of the float at the water-surface the more sensitive the gauge
will be.
To reduce the oscillations of the surface a gauge, whether fixed
or floating, may be placed in a masonry well communicating with
the stream by a narrow vertical slit. It is not certain that the
average water-level in the well is exactly the same as in the
stream, but the difference can only be minute. The larger the
well the better the light, and the less the oscillation of the water.
The advantage of a slit as compared with a number of holes is that
it can always be seen whether the communication is open, but in
order to avoid the necessity for frequent inspection the oscillation
of the water in the well should not be entirely destroyed. In
observations made downstream of the head-gates of irrigation
distributaries in India the oscillations were very violent —
amounting to '60 foot — but they were reduced to '03 foot in the
well by slits -005 foot wide.^
Where a gauge does not exist the water-level can be measured
from the edge of a wall or other fixed point, either above or below
the surface. Owing to the oscillation of the water the end of the
measuring-rod cannot be held exactly at the mean water-level. It
should be held against the fixed point, and the mean reading
taken as explained above. A self-registering gauge can be made
by means of a paper band travelling horizontally and moved by
clock-work and a pencil moving vertically and actuated by a
float. The pencil draws a diagram^ showing the gauge-readings.
The water-level in a tank may be shown by a graduated glass
tube fixed outside the tank and communicating with it.
The level of still water can be observed with extraordinary
accuracy by Boyden's Hook-Gauge, which consists of a graduated
rod with a hook at its lower end. The rod slides in a frame
carrying a fixed vernier, and is worked by a slow-motion screw.
If the hook is submerged, the frame fixed, and the rod moved
upwards, the point of the hook, before emerging, causes a small
capillary elevation of the surface. The rod is then depressed till
the elevation is just visible. By this means the water-level can
bo read to the thousandth of a foot, and even to one five-thousandth
in still water, by a skilled observer in certain lights. The hook-
gauge is not of much use in streams because of the surface
oscillation. It is most used in still water upstream of weirs.
• Gourley and Crinnp usod two 9-inch stoneware pipes placed on end one
above the other. " Also see Notes at end of chapter.
HYDRATILIC OBSEKVATIONS 299
To destroy oscillation and ripples, a box having holes in it may
be placed in the water and the readings taken in the box. When
observing with a hook-gauge in water not perfectly still the
point of the hook should be set so as to be visible half the
time. A pointed plumb-bob hung over the water from a closely
graduated steel tape is sometimes used, and by it the surface-level
can be observed to within -005 foot. The adjustment of the level
of the zero of the gauge above a weir may be effected by a
levelling instrument. If effected from the level of the water
when just beginning to flow over the crest capillary action may
cause some error.
6. Piezometers. — ^The name 'Piezometer,' used chiefly for the
pressure column of a pipe, is also used to include a gauge-well and
its accompanying arrangements. In all such cases the surface,
where the opening is, should be parallel to the direction of flow
and flush with the general boundary of the stream, and the
opening should be at right angles.^ If it is oblique the water-
level in the piezometer will be raised or depressed according
as the opening points upstream or downstream. The well or
pressure tube can be connected with any convenient point by
flexible hose terminating in fixed glass graduated tubes. With
high pressures the piezometers may be connected with columns of
mercury, which may be surrounded by a water-jacket to keep the
temperature nearly constant. Common pressure gauges are not
accurate enough.
In the piezometers of pipes air is somewhat liable to accumulate
and cause erroneous readings. When the presence of air is
suspected the tubes should be allowed to flow freely for a few
minutes. If flexible they can be shaken, and if stiff' rapped with
a hammer. Very small tubes are liable to obstruction by leaves
or deposits and should be avoided, as also should glass gauge-
tubes small enough to be aff'ected by capillarity. The orifices
should be drilled and made carefully flush. Instead of one orifice
there may be four, 90° apart, in one cross-section of a pipe, all
opening into an annular space from which the piezometer tube
opens. It is not certain that this gives greater exactness, but
with a single opening from the top of the pipe the accumulation
of air is probably greatest. The air probably travels along the
pipe at the top.
Pulsations with fluctuation of the water level may occur in
piezometers and should be dealt with as described in art. 5.
1 The sectional area of the pipe at the point of attachment should be the same
as the mean area in the length over which the slope is measured.
300 HYDRAULICS
The arrangements at the weirs where the most important
observations (chap. iv. art. 1) have been made were as below.
In all cases the surface containing the orifice was parallel to the
axis of the stream.
Bazin. — An opening near the bed 4 inches square communicating
with a well.
Francis. — A small box^ with 1-inch holes in the bottom.
Fteley cmd Stearns.- — For the 19-foot weir there was an opening
•04 foot in diameter and '4 feet lower than the crest of the weir.
From the opening a rubber pipe led to a pail below the weir.
For the 5-foot weir there was a board parallel to the side of the
channel and 1 -5 feet from it. The pipe leading to the pail started
from an auger-hole in the board '9 feet above the bed of the
channel.
To find the heads on weirs piezometers connected with perforated tubes
placed horizontally in the channel have been used in America, but they
appear to give unreliable results, even when the holes open vertically. In
experiments made at Cornell University^ the 'middle piezometer' was a
transverse 1-inoh pipe, laid 8 inches above the bed and 10 feet upstream of
the weir. The ' upper piezometer' was similar, but 15 feet further upstream.
A ' flush piezometer ' was also ' set in the bottom of the flume,' 6 inches
upstream of the upper piezometer. The readings of these two diflfered on
one occasion by '3 foot. The readings of the upper and the middle also
difi'ered. It is believed that the opening from the rounded surface of the
pipe, instead of from a plane surface, causes error, and that the error is one
of defect. A ' longitudinal piezometer ' was formed by certain perforated
pipes. With high heads — a little over 3 feet — the longitudinal piezometer
read '099 foot higher than the upper piezometer. With a head of about
■17 foot there was no difference between the two. Experiments made by
Williams ^ also show that the readings obtained with a transverse pipe with
holes opening downwards, do not agree with those obtained by a simple
opening in the side of the channel, being higher with low supplies and
lower with higher supplies. It seems clear that all perforated pipe
arrangements are to be avoided until their action is better understood.
7. Surface-slope. — Probably the best method of observing the
slope in a short length of open stream is to dig two ditches from
the extremities of the slope length, both leading into a well
divided into two by a thin partition. The difference between the
water-levels on the two sides of the partition is the local surface-
fall. It can be very accurately measured, especially if the ditches
' The box projected somewhat into the stream, and this was not free
from objection, as it caused an abrupt change.
'^ Trannactions of the American Society of Civil Engineers, vol. xliv.
'' Ibid. vol. xliv.
HYDKAULIO OBSERVATIONS 301
are treated as gauge-wells, that is, open into the stream by narrow
slits. Slight leakage in the partition is probably of no consequence
as long as it gives rise to no perceptible current in the ditch.
The slope should, unless the stream is perfectly uniform and
straight, be observed at both banks and the mean taken (chap. vii.
art. 16).
For measuring the loss of pressure head in a short length of
pipe or channel a differential gauge consisting of two parallel glass
tubes with a scale fixed between them is commonly used. The two
tubes are connected at the top where there is a cock, and their
lower ends are connected by hose pipes with the two points in the
pipe or channel. Capillarity does not vitiate the results because it
is the difference that is taken. If the tubes are partly filled with
water and the space above the water is occupied by air the
difference in heights of the water columns gives the difference in
head. When this difference would be too small to be accurately
observed, paraffin — specific gravity, say, -80 — can be substituted for
air. It is then as if the specific gravity of the water in the tube
was equal to the difference between the specific gravities of water
and paraifin. The difference in the heights of the two water
columns is five times, more accurately 5'3 times, what it was.
Also see art. 14.
In whatever way slope is observed the openings of any pair of
gauge-wells, ditches, or piezometers must be exactly similar, and
the observations should be repeated at intervals as long as the
velocity observations last.
Section III. — Floats
8. Floats in general. — The size of a float used for point
measurement is limited by the consideration that the mean
velocity of the stream within the ' direct area ' of the float (the
area of its projection on a cross-section of the stream) must be
practically equal to that at the point where the velocity is sought.
The depth of the submerged part of a surface-float may be about
one-twentieth of the depth of water, and the depth of a sub-surface
float one-tenth, or, at the point of maximum velocity, one-
twentieth of the depth of water. The width of a float of any
kind may be about one-twentieth of the width of the stream,
except for use near the bank, when it may be about one-tenth of
the distance from the bank to the line of the float. The length is
302
HYDRAULICS
similarly limited because the float may revolve. The exposed
part of a surface-float should be small compared to the submerged
part. For deep water a good surface-float is made by a bottle
submerged all but the neck, or a log deeply submerged ; for
shallow water by a disc almost totally submerged and carrying a
small cylinder or knob. With all kinds of floats the exposed part
should be of such a colour that it can easily be seen.
The 'lines' or boundaries of the run are marked by ropes
stretched across the stream at right angles, or, if the width is
great, by lines of flags. Observers signal each float as it crosses
the lines, and another observer notes the times. When ropes are
used the float-courses can be marked by ' pendants ' of cloth or
brass. Usually about three floats are signalled in rapid succession
at the first line and then at the second. If on reaching the second
line they have changed order, this aff'ects the individual times
recorded, but not the mean. With a stop-watch the time-
observer may also be the float-observer. He can start and stop
the watch while noting the float. But in this case each float must
complete its course before another can be timed. With a slow
current the time observer may also start the floats, and he may
even use an ordinary watch. In a wide river the course of a float
can be observed by an angular instrument (see art. 3).
A float required to travel in any course usually deviates from
it. The deviation increases the distance over which it travels,
but this is of no consequence because the object is to obtain the
forward velocity (chap. i. art.' 3). The deviation is of conse-
quence only when the velocities in adjacent parts of the stream
differ much from one another, that is, near banks or shallows.
In such cases the 'run' of the float can be shortened, the deviation
noted, and the mean coiirse adopted. When ropes are used the
approximate deviation can be seen by the float-starter by means
of the pendants, especially when the rope is at a low level.
The length over which a float travels, upstream of the run, in
order that it may acquire the velocity of the water, is called the
'dead run.' The float may be taken out into the stream, or
thrown in from the bank, or placed in it from a bridge or boat.
Throwing-in is often practicable with surface-floats, and some-
times with rods. A low-level single-span bridge is the most
suitable .irrangcment, but if there are piers or abutments which
interfere with the stream they disturb the flow, and a site down-
stream of them is unsuitable for velocity measurements, at least
with floats (chap. ii. art. 21). Even a boat causes disturbance
HYDRAULIC OBSERVATIONS 303
downstream. Two small boats or pontoons carrying a platform
are better than a large boat.
The length of run to be adopted depends on the velocity and
uniformity of the stream, the accuracy of the timing, and the
distance of the float-course from the bank, this last consideration
having reference to deviation. Ordinarily the length may be so
fixed that the probable maximum error in timing will be only
a small percentage of the time occupied. The length may,
however, have to be reduced if the stream is not regular, especially
if rods are used. Reduction of the length in order to avoid
excessive deviation is most likely to be necessary for observations
near the bank, especially with surface-floats. The surface-currents
near the bank set towards the centre of the stream (chap. vi.
art. 7), so that the tendency to deviation is greater, while the
admissible deviation is less. Most observations are made at a
distance from the bank, and the rejections for excessive deviation
need not generally be numerous. A moderate number of re-
jections, owing to a long run, does not cause much loss of time,
because in order to obtain a particular degree of approximation
to the average velocity of the stream the number of floats re-
corded must be inversely proportional to the length of the run.
9. Sub-surface Floats. — A float used for measuring the velocity
at a given depth below the surface is called a ' double-float.' A
submerged ' lower float ' somewhat heavier than water, is suspended
by a thin ' cord ' from a ' buoy ' which moves on the surface. In
the ordinary kind of double-float the buoy is made small, and the
velocity of the instrument is assumed to be that of the stream at
a depth represented by the length of the cord, but it is really
different because of the current pressures on the buoy and cord,
and the ' lift ' of the float due to these pressures. There is also
' instability ' of the lower float, caused chiefly by the eddies which
rise from the bed. Any lateral deviation of the lower float adds
to the lift, but otherwise is not of consequence, except near the
banks. The resultant effect of all the faults is a distortion of the
velocity curves obtained. When the maximum velocity is at
the surface (Fig. 112, p. 184) the buoy and cord accelerate
the lower float, and the lift brings it into a part of the stream
where the velocity exceeds that at the assumed depth. Hence
the velocity obtained is always too great, and the 'observation
curve,' which is shown dotted, lies outside the true curve. When
the maximum velocity is below the surface the curve is distorted
as in Fig. 1 1 3.
304 HYDRAULICS
A double-float is best suited to a slow current. The higher the
velocity of the stream the greater the differences among the veloci-
ties at different levels and the greater the lift of the lower float ;
the greater also the strength of the eddies and the instability.
The defects of the double-float cannot be removed, but they can
be much reduced by attention to the design. In order that the
lower float may be as free as possible from instability, its shape
should be such as to afford little hold to upward eddies. In order
that it may be little affected by the current pressures on the buoy
and cord, it should afford a good hold to the horizontal current.
It should therefore consist of vertical plates, say of two cutting
each other at right angles, with smooth surfaces, and lower edges
sharpened. The upper edges should not be sharpened. Any
downward current will then act as a corrective to instability. If
the float tilts much its efficiency is reduced, but tilting can be
prevented by avoiding a high ratio of width to height, and by
making the upper and lower parts respectively of light and heavy
materials, say wood and lead. If the thickness of the plates is
uniform the resistance to tilting is a maximum when the heights
of the heavy and light portions are inversely as the square roots
of the specific gravities of the materials. It is an improvement to
remove the central portions of the plates and to substitute for
them a hollow vertical cylinder, in the middle of which the cord
is attached by a swivel. This causes the pull of the cord, however
the float revolves on its vertical axis, to be applied at the point
where the average horizontal current pressure acts. The cord
should be of the finest wire, and the buoy of light material, say
hollow metal, smooth and spindle-shaped, the cord being att^ached
towards one end, so as to make the float point in the direction of
the resistance.
Given the velocity of the stream the force tending to cause
instability of the lower float depends on its superficial area. Its
stability depends on the ratio of its weight to its superficial area,
that is, on the thickness of the plates. For all floats of the same
shape and materials there is a certain thickness of plate which is
the least consistent with stability, and a float should be composed
of plates of this thickness, in order that the thickness of the cord
and volume of buoy may be small. This thickness cannot be
determined theoretically, but is a matter of judgment and
experience. Of any two similar double-floats, that which has the
larger lower float is the more efficient. If the direct areas of the
lower floats are as 4 and 1, their weights and the submerged
HYDRAULIC OBSERVATIONS
305
volumes of the buoys are as 4 and 1. But the direct areas of the
buoys, if their shapes are similar, are as 4* and 1 or nearly as
2 '5 and 1. The thicknesses and direct areas of the cords are also
theoretically as 2 and 1. In both cases the larger instrument has
greatly the advantage, and practically, if the lower float is small,
it is physically impossible to make the cord thin enough. The
dimensions are limited by the considerations set forth above. The
larger the stream the greater the admissible size of float.
The following statement shows that the double-floats which
have been actually used in important experiments have been of
bad design : —
Channel.
Observer.
Greatest
Depth of
Water.
Description of
Lower Float.
Ratio of Direct Areas
at Maximum Depth.
Lower
Float.
Cord.
Buoy.
Mississippi.
Irrawaddy.
Ganges
Canal.
Humphreys
and Abbott.
Gordon.
Cunningham.
Feet.
110
70
11
Keg with top
and bottom
removed.
Block of
woodloaded
with clay.
Ball (3 inches
and If inch).
1-0
10
1-0
1-75
, -73
{■12)
•03
■06
■10
It is obvious that when the lower float was near the bed — or
supposed to be near it — the observed velocities must, owing to the
very great relative currentactions on the cord, and probably also
to instability, have been so much in excess of the truth as to
render them mere approximations, the general values found for
bed velocities being perhaps about halfway between the real bed
velocity and the mean velocity from the surface to the bed. The
vertical velocity curves obtained with the above instruments often
show marked peculiarities in form, the velocity sometimes seeming
to remain constant or even increase as the bed is approached.
In the ' twin-float ' the submerged part of the buoy or ' upper float ' is of the
same size, shape, and roughness as the lower float, and the velocity of the
instrument is assumed to be a mean between the stream velocities at the
surface and at the level of the lower float. The surface velocity is observed
separately and eliminated. This causes additional trouble. The best form
and size for the lower float are arrived at in the same manner as in the
U
306 HYDRAULICS
ordinary double-float. The difficulties arising from tilting and instability-
can be overcome by making the lower float heavy and the upper one light.
The current pressure on the cord is less than with the ordinary double-float,
but its inclination greater. The instrument has been very little used.
Cunningham has proposed a triple float for measuring the mean
velocity on a vertical when the depth is too great for rod-floats, or
the bed too uneven. It has a small buoy and two large submerged
floats at depths of '21 and '79 respectively of the full depth, the
upper of the two being light and the lower heavy. The instru-
ment is supposed to give the mean of the velocities at these two
depths, and this is nearly equal to the mean on the whole
vertical. The figures '21 and -79 were arrived at theoretically
by Cunningham, and they are the best for general use, the depth
of the line of maximum velocity being supposed to be unknown.
It would be preferable to use a multiple float with several equi-
distant submerged floats, the lower ones heavy and the upper
ones light, the distance of the lowest from the bed and of the
highest from the surface being half the distance between two
adjoining floats. All these floats are best suited to slow currents.
10. Bod-floats. — A rod-float is a cylinder or prism ballasted so
that in still water it floats upright. In flowing water it tilts
because of the differences in the velocities of the stream. By
using a rod reaching nearly to the bed the mean velocity on the
vertical is obtained. Owing to the irregular movements of the
water both the submerged length and the tilt of the rod vary
slightly. The clearance below the bottom of the rod must
be sufficient to prevent the bed being touched. The great advan-
tage of a rod as compared with a multiple float is that there is no
uncertainty as regards lift and instability.
Eods are usually made of wood or tin and weighted with lead.
A wooden rod is liable to alter in weight from absorption of water,
and it may then become too deeply submerged or sink. A cap
containing shot fitted to the lower end of the rod gives a ready
means of adjustment. In a rapid stream a wooden rod may have
an excessive tilt, and a tin rod is better. It is lighter and can
carr more ballast. It is, however, liable to damage and to
spring a leak. A rod may sometimes sink, owing to its ground-
ing and being turned over by the current. In a rapid stream
a wooden rod may bo turned over even without grounding.
Wooden rods can be more easily made square than of other
sections. In any case the section and degree of roughness must
be uniform throughout.
HYDRAULIC OBSERVATIONS 307
For a rod 1 foot long, 1 inch; and for one 10 feet long, 2|
inches are suitable diameters. Rods are often made up in sets, the
lengths increasing by half-feet, or for small depths by quarter-feet,
but this does not give sufficient exactitude, and it often leads to the
use of rods much too short. Owing to the unevenness of the bed
a rod of the proper theoretical length is usually too long, and
the next length is perhaps 10 or 15 per cent, shorter. A set of
short adjusting pieces to screw on to the tops of the rods should
be provided. Rods for use in very deep water are sometimes
made in lengths screwed together. It is convenient to have rods
divided into feet, beginning from the bottom. If the tilt is likely
to be great, allowance can be made for it in selecting the length
to be used.
It has been said that a rod, owing to its not reaching down to
the slowest part of the stream, must move with a velocity greater
than the mean on the whole vertical. Cunningham has attempted
to show theoretically that the length^of a rod must be -945, -927, or
•950 of the full depth of water according as the point of maximum
velocity is at the surface, at one-third depth, or at half-depth.
The proof rests on the assumption that the vertical velocity curve
is a parabola. It has been shown (chap. vi. art. 9) that it is not
a parabola, and that the velocity probably decreases very rapidly
close to the bed, and for this last reason it is probable that a rod
reaching close to the bed would move too slowly. The proper
length of rod cannot be calculated theoretically in the present
state of knowledge.
A large number of experiments with rod-floats were made by
Francis. The discharges obtained by rods in a masonry flume of
rectangular section with a depth of water of 6 feet to 10 feet
were compared with the discharges obtained from a weir in a
thin wall, and the following formula was deduced —
r=F;('l-012--116V:^),
when V is the mean velocity on the vertical, Vr the rod velocity,
d the length ^of the rod, and D the depth of the stream. Accord-
ing to this formula the correct length of rod, so that F'and V,.
may be equal, is •99Z', and the errors due to shortness of rod are
as follows : —
—=•75 •SO -Sb •go •gS ^95 ^96 -97 •QS ^99
D
Z=-9oi -961 -968 -975 -981 •gSG ^989 ^992 -996 1-00
"■ That is, submerged length.
308 HYDRAULICS
The discharges obtained by the weir are believed to be very
nearly correct, and the acceptance of the above figures is recom-
mended. Accepting them, the proper length of a rod is '99 of
the full depth, and if the length is only -93 of the full depth the
velocity found is 2 per cent, in excess. In earthen channels a
rod of the proper length can hardly ever be used, but allowance
can be made for its shortness.
Section IV. — Current-meters
11. General Description. — The current-meter consists of a
screw, resembling that of a ship, and mechanism for recording
the number of its revolutions. Frequently this mechanism is on
the same frame as the screw, and by means of a cord it can be put
in and out of gear. The reading having been noted, the meter is
placed in the water, the recording apparatus brought into gear,
and, after a measured time, put out of gear and a fresh reading
taken. The difference in the readings gives the number of revolu-
tions, and this divided by the time gives the number of revolutions
per second. This again, by the application of a suitable co-efficient,
determined when the instrument is rated, can be converted into
the velocity of the stream. The co-efficient depends on the ' slip
of the screw,' and varies for each instrument and each velocity.
With many meters the recording apparatus is above water, and
there is electric communication between it and the screw. The
meter can then be allowed to run for an indefinite time without
raising to read. For each meter there is a minimum velocity
below which the screw ceases to revolve. This may be as low as
six feet per minute.
Sometimes a current-meter is carried on a vertical pivot and
provided with a vane. The irregularity of the current causes the
instrument to swing about, and so to register the total and not the
' forward ' velocity. It is better to keep the instrument fixed with
the axis parallel to that of the stream, but if the axis swings
through a total angle of 20° — 10° either way — the velocity regis-
tered is only '75 per cent, in excess of the forward velocity, and if
the total angle is 40°, 3 jior cent, in excess.
A current-meter niny bo used in a small stream from the bank
or from a bridge, but generally it is used from a boat. This has
already been referred to (art. 1). The rod or chain to which
the meter is attached should be graduated. If a rod is used, it
HYDRAULIC OBSERVATIONS 309
may be sharpened or rounded on its upstream face, the down-
stream face being flat, and resting against a portion of the platform
fixed at right angles to the centre line of the boat. The rod can
be provided with a collar, which can be clamped on to it in such a
position, that when it rests on the platform the meter is at the
depth required. In water 53 feet deep Revy attached the
meter to a horizontal iron bar, which was lowered by ropes
fastened to its ends, and was kept in position by diagonal ropes.
In shallow water an iron rod is sometimes fixed, on which the
meter slides up and down, but this causes delay.
In some experiments the time in quarter-seconds, position of
the meter, and number of revolutions of the screw have been
automatically recorded on a band driven by clockwork. With a
meter having electric communication with the bank a wire rope
has been stretched across a wide stream, the meter carried on a
frame slung from the rope, and the discharge of the stream thus
observed. In other cases the observers travel in a cage slung
from a wire rope. It is quite usual to have several meters work-
ing simultaneously at different depths. In integration it is not
necessary for the descending and ascending velocities to be equal,
and two or three up and down movements may be made without
raising to read. It is a common practice, after taking an observa-
tion lasting a few minutes, to check it by a shorter one. To
facilitate the computation of the meter velocity the times may be
whole numbers of hundreds of seconds. A stop-watch may be
started and stopped by the same movement, which puts the instru-
ment in and out of gear.
The rate of a current meter is liable, at first, to increase slightly,
owing to the bearings working smoother by use. It should be
allowed to run for some time before being rated. Oil should not
be used, as it is gradually removed by the water, and the rate may
then alter. Every time a meter is used the screw should be spun
round by hand to see that it is working smoothly. A gentle
breeze should keep it revolving. A second instrument should be
kept at hand for comparison. A short test of the rating should
frequently be made. If tests made at two or three velocities
all show small or proportionate changes of one kind similar correc-
tions may be applied to other velocities, but if the changes are
great or irregular the instrument should be rated afresh.
The speed of a current-meter is liable to be affected by weeds,
leaves, etc., becoming entangled in the working parts. If any are
found when the instrument is read the observation can be rejected.
310
HYDRAULICS
but some may become entangled and detached again without being
seen. The effect must be to reduce the velocity, and any abnor-
mally low result may be rejected. The rate of the instrument is
also liable to be affected by silt and grit getting into the working
parts and increasing the friction. The only rubbing surface which
has a high velocity is the axis of the screw, and this is probably
the part chiefiy affected. In using a current-meter of the kind
illustrated (Fig. 144) it was found on one occasion that it rapidly
became stiff. The meter having been cleaned, the screw ran freely
again, but again became stiff. The stream was six feet deep and
had a velocity of about seven feet per second. The water con-
tained silt and probably fine sand, which gradually increased the
friction. The clogging was most rapid in observations below mid-
depth, and it is probable that there was more sand in that part of
the stream.
12. Varieties of Current-meters. — There are probably twenty
kinds of current-meter. Each kind has its own special advantages
or disadvantages. Fig. 144 shows a meter sold by Elliott Brothers,
London. The instrument is attached by the clamping screw to a
rod yi. By pulling the cord D the wheel B is geared with the
screw. A vane F can, if desired, be attached. A meter very
similar to the above is made in the Canal workshops at Koorkee,
HYDRAULIC OBSERVATIONS 311
India, but it is pivoted on the tube which carries the screw for
clamping it to the rod.
In Revy's current-meter friction is reduced by a hollow boss on the axle
of the screw of such a size that the weight of the whole is equal to that of
the water displaced. The recording mechanism is enclosed in a box covered
by a glass plate, filled with clear water, and communicating by a small hole
with the water in the stream, so that the glass may not be broken by the
pressure at great depths. A horizontal vane is added under the screw, so
that it may revolve freely while the meter rests on the bed.
Moore's current-meter consists of a brass cylinder, lOJ inches long, pro-
vided with screw-blades. In front of the cylinder is an ogival head which
is fixed to the frame. The cylinder, which is water-tight, revolves, and the
reading apparatus is inside it, the reading being observed through a pane of
glass. The instrument is hung from a cord or chain. This renders it easier
to manipulate. To prevent its being forced far out of position, a weight is
suspended to the frame, and it should be sufficient to prevent the instrument
being temporarily displaced by the tightening of the gearing cord. The
instrument has horizontal and vertical vanes and can swing in any direction.
In Harlacler's current-meter there is electric connection between
the worm-wheel driven by the screw and a box above water. At
every hundred revolutions of the screw the worm-wheel makes an
electrical contact, and an electro-magnet in the box exposes and
withdraws a coloured disc. The meter slides on a fixed wooden
rod. A tube lying along the rod carries the electric wires, and
serves to adjust the meter on the rod. In one variety the axle
of the screw carries an eccentric which makes an electric contact
every revolution, and thus enables individual revolutions to be
noted.
Fig. 145 shows a current-meter sold by BufiF and Berger,
Boston, U.S.A. The object of the band encircling the screw is
to protect the blades from accidental changes of form, which would
cause a change in the rate of the instrument. A bar underneath
the screw and a stout wire running round at a short distance
outside it aflfbrds additional protection, and enables the instrument
to be used close to the bed or side of a channel. There are two
end bearings and a very light screw and axle, and the screw
revolves with one-fourth of the velocity required to turn a similar
one with the usual sleeve bearing. The friction is so small that
the rate is not altered by silt or grit. The meter is fixed to a
brass tube, which has a line along it to show the direction of the
axis when the meter cannot be seen. The meter is sold with the
recording apparatus either on the frame or with electric con-
nection, as in the figure. Stearns used a meter of this type, and
provided with two screws, either of which could be used. On§
312
HYDRAULICS
(^ natural size.)
Fill. 145,
HYDRAULIC OBSERVATIONS 313
had eight vanes and the other ten In the latter half the vanes
had one pitch and the other half a different pitch. The eight-
vane screw began to move with a velocity of '104, and the
ten-vane screw with a velocity of '094, feet per second.
In the Haskell current meter (Fig. 145a) the screw is somewhat
in the form of a cone with the apex upstream. This shape is
intended to give it strength to resist damage from objects carried
against it, and also to readily throw off weeds. Screws of two
pitches are made. The one with the lower pitch — this appears
to be the more generally used — is suitable for velocities from '2
foot to 10 feet per second, the higher pitch for velocities from
1 foot to 16 feet per second. The bearings are of large surface
and not liable to rapid wear and are under cover, so that grit
cannot affect them. TJie velocity register is above water and has
electric communication with the meter. Starting and stopping
the watch makes and breaks the circuit.
The type meant for use in deep rivers (Fig. 145a) is suspended,
swings on a vertical axis, and is provided with a torpedo-shaped
lead weight. On the Irrawaddy it was found by Samuelson ^ that
a weight of 80 lbs. was necessary. A type for use in small streams,
and made in two sizes, ,is held on a graduated rod. It can be
clamped, or can be left free to swing. A "set-back" velocity
register is also supplied. This can be set back to zero after each
observation. The Ritchie-Haskell "Direction Current Meter"
indicates also the direction of the current, which in a tide-way may
not always be the same as at the surface.
Another well-known current meter is Price's. Both this and
Haskell's are made in the U.S.A.
In the cup pattern of current meter there is no screw. The
wheel is provided with conical cups placed in a circle like the
floats of a water-wheel. Each cup presents its open end to the
stream and is driven downstream. It presents its conical end as
it returns upstream.
Observations made by Groat ^ indicate that in perturbed water
such as a tail race, the results given by a cup current meter may
be 6 per cent, too high, those by a screw meter 1 per cent, too low,
that in violently perturbed water the above differences may be
25 and 3 to 4 per cent, respectively, but that if the meters are
allowed to run long enough the errors disappear. ,
■^ Note on the Irrawaddy River.
2 Proe. Am. Soc. CM., 1912, vol. xxxviii,
314
HYDRAULICS
J^G. 145a.
Fig. I45b (see art. 14)
HYDRAULIC OBSERVATIONS
315
A cup instrument must interfere considerably with the natural
movement of the -wiitcr. In an instrument of the screw type little
more than the edges of the screw, when it is revolving, are presented
to the current.
One kind of current-meter has no regular recording apparatus,
but simply a device for making and breaking circuit and a sounder.
The revolutions are counted by the clicks. A current-meter made
by von Wagner gave its indications by sound, but the counting
wa,s effected by an arrangement like the seconds hand of a watch.
At each stroke, or with high velocities at every fourth stroke, the
observer pressed a button which caused the hand to move one
division.
13. Eating of Current-meters. — The usual method of rating is to
move the instrument through still water with a uniform velocity,
and to repeat the process wdth other velocities covering a wide
range. The instrument may be held at the bow of a boat, or
attached to a car running on rails, or on a suspended wire. In
case the water should not be quite still the runs should be taken
alternately in reverse directions.
lio
ISO
a no
\
\
\
\
\
^^
k i a
Velociiies ITI feet jrer second
Fig. 146.
When rating a meter, the length of run being a fixed quantity,
it is only necessary to record for each observation the time
occupied and the difference of the meter readings. After several
316
HYDRAULICS
observations at nearly equal velocities the entries can be totalled
and averaged.
The following table shows the values of the co-efficients of two
current-meters, and Fig. 146 shows the curves obtained by
plotting them. By means of the curves the co-efficients for
intermediate velocities can be found. It will be noticed that
the co-efficient changes rapidly for low velocities, while for high
velocities it is nearly constant. For moderate velocities the
co-efficients increase and then decrease again, causing a sag in the
curve. The same thing occurred with other meters rated by
Stearns. The cause is not known. The actual values of the
co-efficients depend on the graduations of the reading dials.
Meter sold by Elliott Brothers, London,
Stearns's Meter.
(Fig. 144.)
(Pig, 145.)
Velocities.
Velocities.
Co-efficient
Co-efficient.
Actual.
Meter.
\JKJ~\jM.lX\jX\JlMVt
Actual.
Meter.
•725
■484
1-50
■300
■280
1^070
116
1-05
111
■400
■406
•984
1-81
1-71
106
■500
■550
■909
2-97
2-84
1-05
■750
•862
■870
3-84
3-63
1-05
■950
113
■844
4-71
4-45
106
1-30
159
■823
5-85
5-52
106
1-60
197
•813
6-58
6-21
106
2 00
2-47
■809
2^50
310
■806
300
371
•809
4^00
4-93
■813
500
615
•813
6 00
7 42
•809
7-00
8-69
■806
800
10 00
■800
Certain equations intended to show the law of the variation of the
co-effioient have been arrived at theoretically. The most common is
v=aV+b,
where v is the actual velocity, V the meter velocity, and » and 6 are
constant for any given instrument, their values being selected so as to make
the agreement with the experimental co-efficients as close as possible. The
above is the equation to a straight line, but the co-efficient is given by
c=-^ = a+ — , which is the equation to a curve. The values of the co-
efficients for Stearns's meter calculated by this formula are shown dotted in
Fig. 146. Another equation is
HYDRAULIC OBSERVATIONS 317
Both equations give curves of the same general form, and becoming practically
straight lines at high velocities. They can never agree exactly with curves
having a sag, and as the constants cannot be arrived at until some
experimental co-efficients have been found the equations are not of much
practical value.
It has been shown by Stearns ^ that rating by ordinary towing
through still water is not perfect. In a flowing stream the
velocity and direction of the water constantly vary, but in rating
this is not so. Stearns shows theoretically that the screw turns
more rapidly when the velocity varies than when it is constant,
that an ordinary screw probably turns more rapidly when the
current strikes at an angle than when it is parallel to the axis ;
but that with his meter (Fig. 145) the band and parts of the
frame intercept portions of the oblique currents, and so cause a
decrease in the number of revolutions, the net result depending
chiefly on the design of the instrument. He also moved the
meter with mean velocities ranging up to 3 '7 feet per second
through still water, first with an irregularly varying velocity and
then with its axis inclined to the direction of motion. He found
that inclining the axis 8° and 11° had no appreciable effect, but
that inclinations of 24° and 41° decreased the number of revolu-
tions about 9 per cent.,^ and that with irregular velocities
the number of revolutions was increased, the increase varying
from zero to 5 per cent., being generally greater for low veloci-
ties, and in one case reaching 13 per cent, when the mean velocity
was only -85 feet per second. This velocity was not a very low
one when compared with that for which the screw ceased to
revolve.
By measuring with the same current-meter the discharges in a
masonry conduit, the depths varying from 1 -5 to 4"5 feet, and the
velocities from 1-7 to 2'9 feet per second, and comparing the
results with others known to be practically correct, Stearns found
that, with point measurement, the discharge given by the meter
observations was practically correct, both in the ordinary condition
of the stream and when the water was artificially disturbed, and
that with integration the discharge was correct when the rate
of integration was 5 per cent, of the velocity of the stream, but
too small by 9 per cent, when the rate was 58 per cent, of the
velocity. In the above experiments both the eight-bladed
' Transactions of the American Society of Civil Engineers, vol. xii.
'^ Other experiments have shown that inclinations of 25°, 35°, and 45° give
a decrease in the number of revolutions of 8, 15, and 23 per cent, respectively.
318 HYDKAULICS
and ten-bladed screws were used, the results being generally
similar.
It seems clear that, with the instrument used, the increase in
the velocity due to the variations in the velocity of the stream
was counter-balanced by the decrease due to oblique currents, and
that the instrument gave correct results with point measurements
even when the water was disturbed ; but with an instrument of
different design, and especially one without a band, it seems
probable that the results obtained by point measurement err in
excess, that no additional error is introduced by a moderate
inclination of the axis, or by slow integration, but that rapid
integration causes error. These, however, are only probabilities.
The real lesson to be derived from Stearns's investigations is
that rating effected by steady motion in still water may be
erroneous when applied to running streams, especially with rapid
integration, and that additional tests should be adopted. To
move a meter obliquely or with an irregular velocity would be
troublesome, and would not produce the conditions existing in
streams. It is best to place the meter in a running stream just
below the surface, and to find the velocity by floats submerged to
the same depth as the screw blades. If a sufficient range of
velocities cannot be obtained the meter can be moved upstream
or downstream with a known velocity. This plan can be combined
with ordinary rating. The instrument can also be moved through
still water while giving it a movement as in integration. A com-
parison of discharges obtained by the meter, with results known
to be correct, affords a further test. An immense saving of labour
is obviously effected by rating a number of meters together.
When it is necessary to rely on ordinary rating rapid integra-
tion should be avoided. The error, if any, will probably be less
as the velocity is higher. For ordinary velocities the relative
error is probably nearly constant, so that the results will be
consistent with one another, and sometimes that is all that is
required.
Section V. — Pressxtke Instruments
14. Pitot's Tube. — This instrument usually consists of two
vertical glass tubes o])eii at the ends placed side by side, one the
^pressure tube,' straight, and one the ' impact tube,' with its lower
end bent at right angles and pointing upstream. The water-
level in the pressure tube is nearly the same as that of the stream
HYDRAULIC OBSERVATIONS 319
in whicli the instrument is immersed, but that in the impact tube
is higher by a quantity which is equal to K --, F'being the velocity
of the stream at the end of the tube, and K a co-efficient whose
value has to be found by experiment.
The chief objections to this instrument were originally the
fluctuation of the water-level in the tubes, owing to the irregu-
larity of the velocity, and the difficulty in observing the height of
a small column very close to the water-surface. Darcy in his
gauge tube reduces the fluctuations by making the diameter of
the orifice only 1 -5 millimetres, that of the tube being one centi-
metre. The horizontal part of the tube tapers towards the point,
and this minimises interference with the stream. The difficulty
in reading is surmounted by means of a cock near the lower end
of the instrument, which can be closed by pulling a cord. The
instrument can then be raised and the reading taken. To give
strength and to carry the cock, the lower parts of the tubes are
of copper and are in one piece. For observations at small depths
the heads of the water-columns are in the copper portion of the
instrument, where they cannot be seen. To get over this difliculty
the tops of the tubes are connected by a brass fixing and a stop-
cock to a flexible tube terminating in a mouthpiece. By sucking
the mouthpiece the air-pressure in the tubes is reduced, and both
columns rise by the amount due to the difference between the
atmospheric pressure and that in the tubes, but the difference in
the levels of the two columns is unaltered. The upper cock being
closed and the mouthpiece released, the reading can be taken.
For reading the instrument a brass scale with verniers is fixed
alongside the tubes. The instrument is attached to a vertical rod,
to which it can be clamped at any height, and it can be turned in
a horizontal plane, so that the horizontal part of the impact tube
points upstream. To get rid of the effect of the fluctuations in the
tube several readings, say three maximum and three minimum,
can be taken in succession.
The Pitot tube has been improved by interposing &, flexible hose
between the nozzles and the gauge. The rod carrying the nozzles
is thus more handy and the fluctuations of the water-column can
be watched.
In the Detroit pipe experiments mentioned in chapter v. (art. 4)
the tubes were inserted in the pipes through stuffing-boxes without
interfering with the flow. The diameters of the orifices both
impact and pressure were usually -^-^ inch. When the impact tube
320 HYDKAULICS
was made to point at an angle with the axis of the stream the
reading decreased. When the angle was a little over 45° negative
readings occurred up to an angle of 180°, the greatest negative
reading being for an angle of 90°.
In the Pitot tube the plane of the opening of the impact tube
must be at right angles to the direction of flow. The exact form
of the nozzle is of little consequence. In any case the water in
the tube rises by a height almost exactly equal to — . The chief
difficulty is that the water level in the pressure tube is slightly
different from that due to the pressure. It is usually lower —
because subject to suction owing to the effect of the instrument
on the current— and the co-efficient to be applied to the reading
is then less than I'O. It is usually "80 to I'O, but it may ex-
ceed 1"0.
The practical difSculty with a Pitot tube — as with a current
meter — is the one of rating. Eating in still water may give
results which are wrong by 5 or 10 per cent. The rating should
include tests in a pipe or smooth channel, and the discharge should
be measured in a tank. Tests made by holding the instrument
with its orifice in the centre of a pipe are of course not reliable
because the ratios of central to mean velocity (chap. v. art. 5) are
not sufficiently well known and probably vary not only with D but
with the roughness.
With the Pitot tube no time observations are required. The
instrument is used chiefly in pipes — it can be inserted through
a stuffing-box — and in small channels which are usually smooth.
Parker (Control of Water) considers that it is unreliable in a large
open stream. It is almost certainly unreliable in perturbed water.
Probably it gives best results when the velocity is considerable.
The stream should not be so small that the instrument seriously
obstructs it.
In one pattern of the instrument the nozzles point one upstream
and one downstream, the water in the former being raised and in
the latter depressed. In the I'itnmeter, developed by Coles, the
instrument has upstream and downstream nozzles, as above, and
the two tubes are enclosed in a flat sheath (Fig. 145b, p. 314) and
connected by flexible tubes — not shown in the figure — to a
differential gauge.
There is also an electrically operated device in which the difference
in the pressures in the two tubes is balanced by mercury in a
special form of U-tubo, and equal increments of discharge, in the
HYDKAULIC OBSERVATIONS 321
pipe in which the tubes are inserted, are represented by equal
divisions on a scale. ^ A photographic record of this can be kept.
15. Other Pressure Instruments. —
In Perrodil's Hydrodynamometer a vertical wire carries at its upper end
a, horizontal needle, and at its lower end a horizontal arm, to the end of
which is fixed a vertical disc. The arm ia connected with a graduated
horizontal circle at the level of the needle. When the arm points down-
stream the needle points to zero on the circle. The needle is twisted round
by hand till the arm is forced by the torsion of the wire to a position at
right angles to the current. The pressure of the water on the disc is
proportional to the square of its velocity, and it ia proportional to and
measured by the angle of torsion of the wire as given by the position of
the needle. • The disc oscillates owing to the unsteady motion of the stream,
and the graduated circle oscillates with it, but the mean reading can be
taken. The instrument has not been much used, but it is said to give good
results and to register velocities as low as half an inch per second. It
interferes somewhat with the free movement of any stream in which it is
placed.
The Hydrometrio Pendulum consists of a weight suspended from a string.
The pressure of the current causes the string to become inclined to the
vertical, and the angle of inclination can be read on a, graduated arc.
Except for observations near the surface the current pressure on the string
must affect the reading. Bruning's Tachometer also has an arm and disc,
but the pressure of the water, instead of being measured by the torsion of a
wire, is measured by a weight carried on the arm of a lever. These two
instruments have been little used, and it is not known how far their results
can be relied on.
Section VI. — Pipes
16. The Venturi Meter.^ — The principle of this has been described
in chap. v. art. 7. If D is the diameter of the pipe at the entrance
to the meter, the lengths of the conical parts of the pipe are
generally 2-5 D upstream and 7 5 D downstream, the angle of
divergence in the latter portion being 5° 6'. The area A may be
ia to 18a, but is usually 9a. A has been as great as 60 square
feet and as small as 2 square inches. The opehing from the pipe
into the pressure column may consist of one or more small orifices
or there may be a gap and an annular chamber between the two
portions of conical pipe. The Venturi meter if properly calibrated
is an accurate and trustworthy instrument. It may be inaccurate
with very low velocities unless these have been included in the
calibration. With ordinary velocities c usually ranges in different
instruments from -96 to 1-00, increasing generally with the size of
1 Jowrnal Am. Soc. Mech Eng., vol. xli. ; Engineering Record, vol. xlvii.
X
322 HYDRAULICS
the meter, and for a given meter being nearly independent of the
velocity.
In an investigation by Gibson ^ into peculiarities of the Venturi
meter it is shown that when v is less than about "5 foot per second
c may be as low as '75 or as high as 1"36, and that the instrument
is not reliable for such low velocities unless it has been calibrated
for them ; that — since stream-line motion can occur in the con-
verging cone at velocities much higher than in the main pipe —
when V in the pipe is less than the critical velocity, stream-line
motion may continue up to the throat and that, since in such a case
the kinetic energy of the water is — instead of — , this.may cause
c to be as low as "7. The effects of gaps of different widths were
tested by experiments, and it Was concluded that abnormally high
co-efficients may occur owing to abnormal pressures in the throat
column due to the accumulation of air — when the pressure is
below atmospheric pressure — at the throat, but probably only
when there is a gap at the throat and when the two measuring
columns of the meter are independent. When negative pressures
are anticipated a U-tube gauge should be used and not independent
measuring columns.
17. Pipe Diaphragms. — The discharge of a pipe can be measured
by means of an 'orifice in a thin wall,' the orifice being in
a diaphragm (Fig. 90, p. 136). Holes are bored in the pipe
upstream and downstream of the diaphragm, and pressure
tubes are attached. The difference between the pressures in the
two tubes can be ascertained by means of a U-tube. The pres-
sure in the jet is no doubt a minimum at the most contracted
section, and increases towards CD. This has been proved by
observations by Gaskell^ on a 4-inch pipe, the pressures being
observed at various distances from the diaphragm. Further
observations on pipes whose diameters were about 6 inches and
8 inches show that c,, had very nearly the values given in
chap. V. art. 6, provided the pressures were measured — both
upstream and downstream — not more than 1-5 inches from the
diaphragm. The pressure drop was not high compared to the
pressure. Equation 70 (p. 141) applies to the case of a diaphragm
if a is the area of the contracted stream and if c„ — say -96 to -98
— is substituted for c.
It is convenient here to compare the various formulse for orifices
' Min, Pi-oc. Jvft. O.E., vol. oxoix.
" Ibid., vol. oxovii.
HYDKAULIG OBSERVATIONS 323
when there is velocity of approach. Let n (equation 8, p. 13) be
taken as 10. In equation 23 (p. 48) let A--=ma = Ma. Then,
since c = c^c,. and a = c^a,
/ 1 / M^~
. (82a).
*-»
In equation 70 (p. 141) let H be put for H—h and c^ for c.
There is no contraction. Then
Q = c,AV2gS^^^ . . . (82b).
Since a is here the minimum area, m is the same as M in
equation 82a. The two equations are for practical purposes
identical. The slight difference is due to c„ being introduced at
the beginning of the working leading up to equation 23. For pipe
diaphragms Gaskell gives an approximate formula
Q^-eOAVigH^^^^ ■ ■ ■ (82c).
In this case there is of course contraction. The results, calculated
for orifices of various diameters (d), obtained from equations 82b
and 82c agree very closely, as long as — is not less than 2.
One kind of diaphragm used by Gaskell was "56 inch thick — the
downstream side of the orifice being bevelled so as to make a thin-
wall orifice — and the holes for the pressure tubes were drilled into
it radially from the outside and ran into holes, one of which opened
upstream and one downstream. For experimental work tRis kind
of diaphragm was not suitable, because whenever the diaphragm
was changed the pressure tubes had to be disconnected. A thin-
plate diaphragm was therefore used, and the holes were drilled in
the pipe flanges, which were sufficiently thick.
Observations on diaphragms in a 5-inch pipe have been made by-
Judd.^ The maximum drop in pressure from the upstream to tha
downstream side of the diaphragm was that due to a head of about
6 feet of water. Of this the percentage recovered was about 77'
when the diameter (d) of the orifice was 'dl}, 30 when d was -dBy
and 4 when d was -22). The recovery had always ceased at a point
distant 4Z> from the diaphragm. On the upstream side of tha
1 Trans. Am. Soc. Mech. Eng., 1916.
324 HYDRAULICS
diaphragm the pressure usually fell — but very slightly — in going
upstream, but became constant long before a distance of one pipe-
diameter was reached. Judd observed some pressures very near
the diaphragm and some further away. The pressure drop through
the orifice was often high compared with the actual pressure. The
co-efficients are somewhat irregular, but on the whole confirm
Gaskell's equation 82o above.
If the pressure is observed so far downstream as CD (Fig. 90),
the pressure drop may not be sufficient to give accurate restdts.
If observed — as in the cases of both sets of experiments above-
mentioned— nearer the diaphragm, the pressure observed is that in
the eddy. This pressure probably differs slightly from that in the
jet, but this need not prevent complete and reliable sets of co-effi-
cients being obtained by further experiment. Uniformity of pro-
cedure is desirable. It seems suitable to observe the downstream
pressure at a point opposite the contracted section. Upstream of
the diaphragm the pressure in the eddy is — ^judging from the case
of a weir (chap. iv. art. 4) — greater than in the actual stream.
The pressure should be observed clear of the eddy, say 'bD to \D
upstream of the diaphragm.
A diaphragm is vastly less costly and easier to instal than a
Venturi meter. At present it is not so accurate. It causes more
loss of head.
The lower ends of Judd's experimental pipes were fitted with caps
having orifices in them of the same sizes as the diaphragm orifices.
The co-efficients obtained in these and some previous experiments
with 3-inch and 4-inch pipes are somewhat irregular, and when
averaged slightly in excess of those obtained for diaphragms.
Judd made some experiments with eccentric and segmental
orifices.* These are not dealt with above. It would seem to be
desirable to attend first to the ordinary concentric orifices and
obtain reliable co-efficients for these.
Notes to Chapter VIII
Self-recordmg Oauge (art. 5). — The float can be made to turn a
drum which, provided with a screw thread of varying pitch and
with simple mechanism, causes the pencil to move equal distances
for equal inurements in the discharge, and such distances can
be magnified {Journal Am. Soc. Mech. Eng., 1912, vol. 34).
Floats. — A rising float consists of a hollow ball, say of copper.
It is held down on the bed of the stream and released at a given
HYDRAULIC OBSERVATIONS 325
moment. Its position on reaching the surface is noted and the
distance of this point downstream from the point of release gives
the horizontal distance travelled by the float. The time taken is
independent of the velocity and depends only on the depth, and
can be ascertained beforehand, so that no time observations on the
spot are needed. The arrangement is suitable for slow currents
where current meters would not be reliable. In case the point of
release cannot be exactly located, two balls of different specific
gravities can be used and the difference between their points of
emergence noted.
CHAPTER IX
UNSTEADY FLOW
Section I. — Flow from Orifices
1. Head uniformly varying. — Let the head over an orifice
during a time t vary from JS^ to H^, and let the discharge in this
time be Q. The mean head or equivalent head H' is that which
would, if maintained constant during the time t, give the dis-
charge Q. Let the head R vary uniformly, that is, by equal
amounts in equal times, as, for instance, in the case of an orifice in
the side of an open stream, whose surface is falling or rising at a
uniform rate. In this case h=Ct where is constant. Let a
be the area of the orifice and c the co-efficient of discharge, which
is supposed constant. The discharge in the short time dt under
the head h is
dQ=ca 'J^gh dt=ca ij'2g0 1^ dt.
The discharge between the times T^ and T^ is
f"' — —
Q=j ca j2gG i^ dt=^ca j2gC (rj- r,f )
=^caj2gC^^^.
Under a fixed head H'
Q=caj2^' {T,-T,) = ca J^JIf ^~~^-
Equating the two values of Q
V//=.|^_-«^. . . (83).
If lii = 0, that is, if the head varies uniformly from Hi to or
from to II „
JII =°f^lH, . . . (84),
or the equivalent head is — '.
2. Filling or Emptying of Vessels. — Let water flow from an
326
UNSTEADY FLOW 327
orifice in a prismatic or cylindrical vessel whose horizontal
sectional area is A. The discharge in time dt is d Q=A dh=ca
'J2gh dt :
,,_ J dh __A h'^dh
The time occupied in the fall of the surface from JI^ to H.^ is
/= {'J: A-V dh= 2-.^__{i/-,i-/f,i).
Under a fixed head H'
f_ A(II,-I I,)
caJ-2gH'
Therefore J IT = -Mi:zEl__ . . . (85).
This is useful for canal locks.
If 11^ = 0, that is, if the vessel is emptied down to the level of the
orifice,
\
JR^-lEl . . . (86).
"S
5
T
The following are the ratios of t^-ff' to ^JHl for certain cases : —
For a prism or cylinder, ....
For a sphere, ......
For a hemisphere concave downwards, .
For a hemisphere concave upwards,
For a cone with apex downwards, .
For a cone with apex upwards,
For a wedge with point downwards.
For a wedge with point upwards, .
For a vessel whose vertical section is a "parabola with
vertex downwards : —
When all vertical sections are the same, . . f
( Paraboloid of revolution).
When the horizontal sections are rectangles, . . §
(Two opposite sides of the vessel rectangles and two parabolas).
In the last case the surface falls at a uniform rate as in the case considered
in art. 1.
In all cases the times occupied in emptying the vessels are
greater than with a constant head i?i, in the inverse ratios of the
above fractions. If a vessel is filled, through an orifice in its
bottom, from a tank in which the water remains level with the
top of the vessel, the ratio of JE' to ^JH^ is the same as for filling
the vessel when inverted. Thus for a cylinder, prism, or sphere
the time for filling is the same as for emptying.
328
HYDRAULICS
If two prismatic vessels communicate by an orifice, and Hi is
the diiference in the water-levels of the vessels, and A^ and J.^
their horizontal areas, the time which elapses before the two heads
become equal is
^_ 2Ai J, J'Hi
caj2^{Ai+A,)
and is the same whichever is the discharging vessel. This
equation may be used for double locks
■ (87),
Section II. — Flow in Open Channels
3. Simple Wayes. — Let ABG (Fig. 147) represent the surface of
a uniform stream in steady flow, the reach commencing from a fall
Fio. 147.
over which is introduced an additional steady supply q, such
that the surface will eventually be EF. A wave is formed below
A, the surface assuming successively the forms GS, G'H", etc.
The point H travels downstream at first with a very high velocity
— since the slope GH cannot remain steep for any but an extremely
short time — but its velocity decreases as the slope at JT becomes
less. The point G rises at a continually decreasing rate, because in
equal times the volumes of water represented by GG'ITH, etc.,
are equal. Obviously the velocity of the point H is greater as q
is greater, that is, it depends on the amount of the eventual rise.
It must not be supposed that the actual velocity of the stream
even at its surface, or velocity of 'translation,' is anything unusual.
As in other cases of wave motion it is the form of the surface
which changes rapidly.
When the surface has risen to E the wave advances only down-
stream, and there is formed a reach EK, in which the flow is
steady and uniform. On consideration it will be seen that if the
UNSTEADY FLOW 329
channel is long enough, the elongation of the wave ceases, its
profile KC becomes fixed, and it progresses at the same rate as the
mean velocity in the risen stream EK. The motion of such a wave
is uniform, and the mean velocity of the stream is the same at all
cross-sections. The proof given in chapter ii. (art. 9) applies to
any short portion of the wave. The pressure on the upstream end
is greater than on the downstream end, but the surface-slope is
greater than the bed-slope, and the equation comes out exactly the
same, S being the surface-slope. At different cross-sections in the
wave S is greater as B is less, so that V is the same everywhere.
Obviously the wave is convex upwards. If at any cross-section in
the wave the slope were less than that required by the above con-
sideration, the velocity there would be reduced, the upstream
water would overtake it and increase the slope. If the slope at
any cross-section were too great, the velocity there would be in-
creased, and the water would draw away from that upstream of it.
Thus the wave is in a condition of stable equilibrium, and always
tends to recover its form, should this be accidentally disturbed.
The curve KC produced to M and N gives the profile of the wave,
supposing the original water-surface to have been DM, or the
channel to have been dry.
Thus the flood-wave has two distinct characters according as its
profile is forming or formed. The forming wave rises as well as
progresses, its velocity is at first very high, and it depends on the
amount of the rise that is on the height AE. The formed wave
progresses at a uniform rate, and its velocity depends only on that
of the risen stream, and not on the amount of the rise. The
surface is in all cases convex upwards. Since any change in the
form of the wave occurring at either end would be communicated
to the whole of it, it is probable that, in ordinary cases, the
moment of time when the point R commences to move with a
uniform velocity coincides nearly with the moment when the
point G ceases to rise, or the wave becomes formed.
As to the form of the curve KG, the case is analogous to that of the sur-
face-curve in variable steady flow (chap. vii. art. 13). The slope at L is
such as will, with uniform flow and depth LP, give the same velocity as the
depth KR with slope EK. Thus the surface-slope corresponding to any
depth is known, and tangents to the curve can be drawn, but the distance
between two points where the depths are given is not known. In a case of
steady flow, with a drawing down KB, the surface-slope at L must be
greater than in the wave now under consideration, because in that case V
is greater than at K instead of being the same, and also because Fis con-
tinually increasing and work being stored.
330 HYDRAULICS
In the case of a reduced steady supply at S (Fig. 148) the sur-
face assumes the forms ST, S'T, etc., the point T travelling with a
PlO. 148.
decreasing velocity and S falling with a decreasing velocity. The
surface eventually assumes the form VZ W, the portion VZ being
in uniform flow. If the original surface is ?7Fthe curve is ZWY.
The velocity in ZWis. lower than in WN, so that WN continually
draws away while ZW lengih&as, and flattens. The angle at W is
no doubt rounded off, so that there is a wave-like form.
Ordinarily the curve of a wave is of great length, and the con-
vexity or concavity slight. If the point L is such that the volumes
KFL and LQC are equal, the time at which this point in the wave
will reach any place, after the wave is formed, is found by divid-
ing the distance of the place from E by the velocity of the risen
stream.
If the additiona,l supply introduced, or the supply abstracted,
instead of being steady, is supposed to change gradually as
would be the case if it were caused by a wave coming down the
upper reach or by the opening or closing of gates or shutters, the
wave below A or X does not at its commencement travel with
such rapidity, and it more quickly assumes its fixed form, unless
the water is introduced or abstracted too slowly to allow it to
do so.
The form of a flood-wave may be observed by means of a
number of gauges, but the wave, except when it is first formed —
and even then if the change in the supply is not made with great
abruptness — is of great length, and its form, or even the times of
passage of its downstream end, can be accurately found only by
very exact gauge readings. Slight changes in the supply, owing
to rainfall or similar causes, are sufliicient to vitiate the observa-
tions. Absorption of water by the channel, especially in the case
of a wave travelling down a channel previously dry, may also
UNSTEADY FLOW 331
greatly affect the movement and form of the wave. On the
Western Jumna Canal in India, with a mean depth of water of
about 7 feet, and a velocity of about 3 -5 feet per second, a rise or
fall in the surface of -25 foot to '55 foot, caused by the manipula-
tion of regulating apparatus, and occupying in each case less than
an hour, was found to occupy 5 or 6 hours at a point 12 miles
downstream, and 6 to 7 hours at a point 40 hiiles downstream.
Attempts made to observe the form of the wave failed owing to
the causes just mentioned.
4. Complex Cases. — Let a rise be quickly succeeded by a fall
(Fig. 148a). As ZTT flattens, the point TF overtakes P. The wave
PC, no longer having behind it the
steady stream WP, also flattens
and the velocity of C decreases.
The whole wave flattens and its
velocity continually decreases.
If a fall is quickly succeeded
by a rise the wave overtakes the p , ■„
trough. But it cannot fill it up.
This would imply that the discharge passing a place lower down
was the same as if no temporary diminution had occurred. The
wave, as soon as it overtakes the other, begins to rise on it, suffers
a decrease of slope, and is checked while the front wave receives an
increase of slope and is accelerated. The trough lengthens in-
definitely. At places a long way down the fluctuations in the
water-level are slight in amount but long in duration.
Given the height of a flood at A (Fig. 147), the full effect of
the flood will be felt at any place K only when the height at A is
maintained for a sufficiently long period. If this period is pro-
longed indefinitely the rise at K will not be increased, except in so
far as may be due to the cessation of absorption by the flooded
soil, but if the period is shortened the rise at K may be greatly
reduced. Empirical formulae intended to give the height of a
flood at any place, in terms of the heights in some reach upstream
of it, must include the time as a factor, or, what is probably a
better plan, must include gauge readings at several places up-
stream, and not at one place only. This plan has been adopted on
various rivers, the places selected being generally those where
tributaries enter. Sometimes it is sufficient merely to add to-
gether the different readings and take a given proportion.
If the channel is not uniform the form of the wave, even if it
has once become fixed, changes. At a reduction of slope the
wave assumes a more elongated, and, at an increase of slope, a
332 HYDRAULICS
more compact form. At an increase of surface-width, supposing
the mean velocity to be unaltered, the wave is checked because
additional space has to be iilled up. At a decrease of width the
velocity of the wave increases.
When an additional supply is introduced or abstracted at a
place where there is not a fall, the water-surface upstream is
headed up or drawn down, and the form which it eventually
assumes may be found hy the methods explained in chapter vii.
(art. 13). The volume of water eventually added to the stream
upstream of the point of change can thus be found, but the
time in which it is added cannot easily be found, because it is
not known how much of the supply passes downstream. The
commonest case of the kind is that of the tide at the mouth of a
river. When the tide begins to rise the water in the river is
headed up and its velocity reduced. As the rise of the tide
becomes more rapid the discharge of the river is insufficient to
keep the channel filled up so as to keep pace with the rise of the
tide, the water in the mouth of the river becomes first still and
level, and then takes a slope away from the sea and flows
landwards. At a place some way inland the water-surface forms
a hollow and water flows in from both directions. This may
obviously continue for some time after the tide has turned, and
high-water then occurs later at the inland place than at the
mouth of the river, a fact which " is sometimes unnecessarily
ascribed to 'momentum.' A sudden and high flood in the Indus
once caused a backward flow up the Cabul River where it joins
the Indus.
If in a long reach of a river the flood water-way is reduced (say
by embankments which prevent flood-spill, or by training-walls
which cause the channel inside them to silt up) a flood of any kind
will, in most of that reach, rise higher and travel more quickly
than before. The same effect will be produced, but to a less
degree, at places further downstream. When the rise is followed
by a fall the wave will not flatten out to the same extent as
before. In the case of a permanent rise, jxcept in so far as there
will have been less absorption than before in the flooded area,
matters will be as before.
5. Remarks. — Sometimes a wave motion is seen in a stream
at some abrupt change where air, becoming imprisoned, escapes
at intervals.^ (Cf. unstable conditions at weirs, chap. iv. arts. 10
and 13.) It is believed that in a falling stream the surface is
' In flow through a bridge the water surface may rise in a wave in one span
while it lulls in the other and vice versd, the moveniont continuing rhythmically.
UNSTEADY FLOW 333
slightly condave across, and in a rising stream convex, but the
curvature is extremely small.
The action of an unsteady stream on its channel is, no doubt,
subject to the same laws as in a steady stream. At the front end
of a rising wave the relation of V to D is exceptionally high, and
scour is likely to occur. At the advancing end of a falling wave
the reverse is the case, and hence a falling flood frequently causes
deposits. In discussions on the training of estuaries the idea has
often been put forward.as a general law that it is wrong to diminish
the flow of tidal water. No doubt it is the tidal water which has
made the estuary. If only the upland water flowed through it
the size would be far too great for the volume. The salt water
may enter an estuary comparatively clear and return to sea silt-
laden. But if training-walls are made so as to reduce the volume
of tidal water entering the estuary, the width to be kept open is
also reduced. No such sweeping law as that above stated can be
upheld. The Thames embankments in London contracted the
channel and to some extent interfered with the tidal flow, but the
channel was scoured and improved.
If a stream is temporarily obstructed by gates, and the water
headed up, the silt deposited, if any, is removed again when the
gates are opened. The same is true of obstruction caused by the
rise of tides. If a given volume of water is available for the.
flushing of a sewer, it can probably be utilised best by introducing
it intermittently, suddenly, and in considerable volumes at various
points in the course of the sewer, commencing from near the tail
and proceeding upwards. If there are any falls or gates it is
clearly best to introduce it just below a fall or below a closed
gate.
Ordinarily, in a rising or falling stream, the relative velocities
at different points in a cross-section are probably normal or nearly
so, but where the fresh water of a river meets the sea the relations
are apt to be much disturbed, especially near the turns of the tide.
The fresh water, being lighter, may rise on the salt water, which
may have a movement landwards, while the fresh water above
it is moving seawards. Such a landward current is obviously not
the result of the surface-slope, and must be due to momentum and
hence temporary. Even where the water is all fresh the relative
velocities may be disturbed. At the turn of the tide the surface
water may begin to move before the lower water.
CHAPTER X
DYNAMIC EFFECT OF FLOWING WATER
Section I. — General Information
• 1. Preliminary Remarks. — Hitherto we have been concerned
almost entirely with questions relating to velocities, discharges,
and water-levels. In this chapter will be considered questions
relating to the Dynamic Effects of Flowing Water. In all cases
the effect of friction will be neglected.
By dynamic pressure is meant the pressure produced by a
stream of water when its velocity or its direction of motion is
altered. This is, of course, entirely different from static pressure.
Let V, A, and Q be the velocity, sectional area, and discharge of
a stream, and W the weight of one cubic foot of the liquid. The
volume discharged per second is A V, and its momentum is
WA — The force which, acting for one second, will produce or
destroy this momentum is F=WA — . On this principle the
pressures developed in various practical cases can be ascertained.
Before proceeding to them it will be convenient to give two
theorems regarding currents, though these do not strictly fall
under the heading of this chapter, and might have been given in
chapter ii. if they had been required sooner.
2. Radiating and Circular Currents. — Suppose water to be
supplied by the pipe AB (Fig. 149), and then to flow out radially
between two parallel horizontal surfaces GD and EF, whose dis-
tance apart is d. Of radii 7?„ i?„, let i?j be the greater, and let
the velocities be F,, Fj, and the pressures /'„ P^. Since the
discharges past all vertical cylindrical sections are equal, therefore
' = = ' . Also since liy Bernouilli's theorem the hydrostatic head
"- 1(^+ 2g- >r-t- ig - w^b;^ 2g •
884
Therefore
And
DYNAMIC EFFECT OF FLOWING WATER
v:-
335
P.
--II -
P V '
2;/
7> 2
Ho
or the heights in pressure columns increase from the centre out-
wards and tend to reach, though never reaching, the value H. If
K
> I
I
I '' I '
A
Fia. 149.
the water flows inwards and passes away by the pipe the law is
the same. A curve through the points G, H, K, etc., is known as
Barlow's curve.
In a vessel (Fig. 150) which, with its contents, is revolving about
a vertical axis with angular velocity a, the
forces acting on a particle A whose velocity
is u are its weight wor AC, acting vertically,
and a horizontal centrifugal force w — or
gx
2
W-—X or AB. The water-surface takes a
form normal to the resultant AD of
the above, that is, the angle D AC is tan~^
Fio. 150.
TT dy a'
Hence -j^=—x.
ax g
Integrating, y=--x^, or the curve EA is a parabola with apex
at E. Since u=ax, therefore y-=—, or the elevation of any point
above E is the head due to its velocity of revolution. The
theoretical velocity of efflux from an orifice at i^* or i? is that
due to a head AF or GB.
A similar condition occurs in a mass of water driven round by
radiating paddles. In either case the condition is termed a ' forced
vortex.' Questions connected with the pressure in a radiating
336
HYDRAULICS
current or in a forced vortex enter, though not to a very impor-
tant degree, into the theories of certain hydraulic machines. In a
centrifugal pump the pressures in the pump-wheel follow the law
of the radiating current, while those in the whirling chamber out-
side the wheel depend on the law of the forced vortex.
u
A. C
n
Section II. — Reaction and Impact
3. Beaction. — Let a jet issue without contraction from an
orifice A (Fig. 151) in the side of a tank. The force i^ causing
the flow is the pressure on B. This
force is called the reaction of the jet.
It tends to move the tank in the
direction AB. It is equal to WA — ,
9
or to 2WAH where H is the head
due to V. If the tank is supposed
to move with velocity v in the direc-
tion AB, the absolute velocity of the
issuing jet is V—v, but the quantity
P;(, 151 * issuing is still AV. Hence the
momentum of the discharge per
second is WA — '.
9
The principle of reaction has been utilised in driving a ship,
water being pumped into the ship and driven out again stern-
wards. The energy of the water just after leaving the ship is
jVAvi^^:^.
The work done on the ship is
Fv=WA^l=l''> . .
9
The total" work done on the water is the sum of the above or
WAV'~-1- . . . (89).
2g
The efficiency of the machine is the ratio of (88) to (89) or
(88).
V+v
The nearer v approaches V the nearer the efficiency is to 1 -0, but
the less the actual work done on the ship. If F=v the efficiency
is 1 0, but the work done is nil. In the IFaterwitch V was 2v,
so that the efficiency was |.
The principal of reaction has also been applied in driving a
DYNAMIC EFFECT OF FLOWING WATER
337
'Reaction Wheel or 'Barker's Mill '(Fig. 152). The preceding
formulae and remarks apply to this case, v being the velocity of
the rotating orifices. If ^C'is the head in
the shaft the head over the orifice D is
BD, AB being an imaginary water-surface
found by the principles of article 2. If
AC=H the velocity of efflux at D is
J'igH+v\
4. Impact. — When a jet of water (Fig.
153) meets a solid surface which is at rest,
it spreads out over the surface. There is
not, strictly speaking, any shock, but there
is loss of head owing to abrupt change.
If the surface is horizontal and a jet strikes
it vertically, it spreads out equally in all
directions. In other cases the amount and
directions of spreading depend on the
circumstances. In all cases, without excep-
tion, the velocity of the jet relatively to
the surface is the. same after impact as ^^^ j^g
before. The flow after impact is along the
surface which, being smooth, cannot alter the velocity of the
water, but only force it to change its direction. The pressure
between the fluid and the surface in any
direction is equal to the change of
momentum in that direction of so much
fluid as reaches the surface in one second.
Let a jet AC (Fig. 151) meet a fixed
plane surface at right angles. The
momentum in the direction ^C is wholly
destroyed and the pressure on the plane
is WA — , or the same as the pressure
Fig. 153.
g
(reaction) on B, or twice the pressure due
to the hydrostatic head which produces V. Thus the pressure on
DE will balance the pressure due to the head FG where FO is twice
KB. In the case shown in Fig. 97 (p. 141) the two heads are
equal. In that case the head HG has to be produced, the discharge
rising through GH. In the present case the head FG has merely
to be maintained.
If the plane is moving with velocity v in the same direction as
the jet the discharge meeting the plane per second is A{V—v) and
338
HYDRAULICS
IV— vY
the pressure is JVA '-. The work done on the plane per
second is WA
{V-vY
The total energy of the water before
V- liV—vYv
impact IS IFAJ-^. The efficiency is— ^^ — ^^ . This is a maxi-
mum when J'=3v and the efficiency is then /y.
If for the vane there is substituted a series of vanes, as in the
case of a jet directed against a series of radial vanes of a large
wheel, the discharge reaching the vanes per second is AV and the
whole pressure is WA V^ — ZJH, The work Bone per second is
IS — ^„, ^ or 2«)- -.. It IS a
F
maximum when «=-„-, and is then -|.
If the vane is cup-sha'ped (Fig. 154), so that the water leaving
the vane is reversed in direction, the velocity of the water leaving
the vane has relatively to the Vane a velocity F—v in a backward
(F—v)v
WA F- and the efficiency is — ^pw
FlG. IB-l.
Fig. 155.
direction aiid an absolute velocity v—J"+v or 2i'—F. The
change of momentum per second is ff'A^-^''{r—{2v—T^} or
2 JVA - — , and the pressure on the cup is double that on
the plane considered above. The work done on the cup is
'2fVA ~ --' V. The efficiency is -^ ~^''. It is a maximum
when V=2v, and is then |. In the case represented by Fig. 155
vha pressure on the solid MN is double that due to a single
cup.
DYNAMIC EFFECT OF FLOWING WATER 339
If there is a series of cups the discharge per second reaching
V
them is AV the whole pressure is WA—{V—{2v—V)] or
V(V—v) iF(V-v)v
2WA-^ -. The efficiency is — ^p^^ — ~. It is a maximum
when V= 2«, and is then 1 '0.
The preceding cases illustrate the great principle to be adopted
in the design of water-motors such as turbines and Poncelet wheels,
namely, that the water shall leave the machines deprived, as far as
possible, of its absolute velocity. If it has on departure any
velocity it carries away work with it. In the' last case it had no
velocity and the efficiency is 1 '0.
Another principle is that the water shall impinge on the vane
so as to create as little disturbance as possible — that is, as nearly
as possible tangentially to the vane — and thus minimise loss of
energy by shock. When the jet strikes tangentially it has no
tendency to spread out laterally, but slides along the vane. In
practice an exact tangential direction is impracticable, but the
vanes are provided with raised edges which prevent lateral spread
and cause the water to be deflected entirely in one plane.
A third principle is that all passages for water shall, as far as
possible, be free from abrupt changes in section or direction, so
that loss of liead from shock shall be avoided.
Let AA' (Fig. 156) be a surface or vane moving in the direction
Pia. 166.
and with the velocity v, represented by Av, and let A V represent
the direction and velocity F oi a. jet impinging on the vane. Let
340
HYDRAULICS
a be the angle between the two lines. The line i)F represents the
velocity V of the jet relatively to the vane at A. Let it be
assumed that the jet is deviated entirely in planes parallel to the
figure. The jet leaves the vane at A' with the velocity V,
represented by the line A'E'. Draw A'v' equal and parallel to
Av. Then A'u represents the absolute velocity of the water
leaving the vane. Let the angle v'A'u=9 and BA'E'=/3. If the
quantity of water reaching the vane per second is w, the original
and final momenta of the water resolved in a direction parallel
10 w
to Av are - V cos n and - V cos 6. The change of momentum
w
or pressure in the direction Av is — ( ^ cos a—V cos 6) or
w
— (i^cos a—v-\-V' cos /8). These are general expressions cover-
ing all cases, and the preceding ones can be derived from them.i
When a jet impinges on a plane, as in Fig. 157, the issuing
velocity of the jet is theo-
H-,
jr
retically JigH^, but on reach-
ing the plane the velocity V
is about JigH. The outer
streams at A press on the
inner by reason of centrifugal
force, and the intensity of
pressure increases towards the
centre of the jet. It cannot
exceed the amount due to —
2<7
or R, because otherwise the
direction of flow would be
reversed. Experiments made
by Beresford - with jets -475
inch to 1'95 inch in diameter falling on a brass plate show
Fm. 157.
^ Some machines which illustrate the principles of dynamic pressure
have been referred to above. There are many machines such as water-
meters, modules, rams, presses, pumps, water-wheels, and water-pressure
engines which, though water passes through them, illustrate no principle of
hydraulics, the questions involved in their design being engineering and
dynamical. In fact, the principles involved in the above formulse regarding
vanes are dynamical, and are given here to bridge over a gap between
hydraulics and another science. The same remark applies to parts of the
succeeding artinlc.
° Professional Papers on Indian Engineering, No. cccxxii.
DYNAMIC EFFECT OF FLOWING WATEll 341
that, at the axis of the jet, the pressure is very nearly that
due to H, and the pressure becomes negligible at a distance
from the axis equal to about twice the diameter of the jet. The
pressure is thus distributed over an area of about four times that
of the section of the jet. The pressures were measured by means
of a water-column communicating with a small hole in the plate
whose position could be altered.
5. Miscellaneous Cases. — When water flows round a bend in
a channel the dynamic pressure produced on the channel is the
same as if the channel was a curved vane. At bends in large
pipes anchors are sometimes required to hold the pipe.
When a mass of water flowing in a pipe is abruptly brought to rest
by the closure of a gate or valve the pressure produced \%f=y^— j.rrn
where L is the length of the pipe affected by the pulsation, m and M the
moduli of elasticity for water and for the material of the pipe in pounds
per square inch, T the thickness of the pipe in inches, r the radius of the
pipe in feet, and v the velocity of the water in feet per second, / being in
pounds per square inch over and above the static pressure.^
When a thin plate (Fig. 158) is moved normally through still
water with velocity V, a mass of
water in front of the plate is put
in motion, and those portions of it -'"^'-''^
which flow off at the sides of the -T's^^ $
plate cannot turn sharp round "^l'^^-
and fill up the space behind ^^l',--
the plate. Instead of doing this ^la. iss.
they penetrate into the rest of
the water and so communicate forward momentum to it, while
other portions of still water have to be set in motion to fill up
the space behind. Thus there is produced a resistance which is
independent of friction or viscosity. Practically it is found that
.,-.-, the resistance is KWA —
~. 2^
>;-- where K is 1-2 to 1-8,
.-^"^ the best results giving
•>'' 1-3 to 1-6. The resist-
^----' ance is less than that
Fig. 159. caused by the impinging
on a fixed plane of a jet
of the same section as the area of the plate with a velocity /''
' ilin. Proc. Inst. G.E., vol. oxxx.
342 HYDRAULICS
If for the plate there is substituted a cylinder (Fig. 159) whose
length is not more than about three diameters, the resistance is
less than in the case of the plate. It is further reduced if the
downstream end of the cylinder is pointed,^
In the above cases, if the plane or cylinder is fitted and the
water moving, the pressures are the same.
The following statement shows the approximate results of some
experiments made by Hagen to show the position assumed by a
rectangular plane surface when pivoted (Fig. 160) and placed in
flowing water ; —
-=l-0' -9 -8 -7 -6 -5 -4 -3 -2
y
^=90° 74° 59° 46° 27° 13° 7° 6° 4°
When a thin sharpened plate or a spindle-shaped or ship-shaped body
is moved endways through still water the resistance is almost wholly
friotional and is nearly as V^, but if the body is only partly submerged
waves are produced, and when V exceeds a certain limit (which bears a
relation to the size of the body) the wave resistance increases and the total
resistance increases faster than F^. If the body, though sharp at both ends,
tapers more rapidly at one end than at the other, it probably causes least
resistance when the blunter end is forward.
In experiments made by Froude by towing boards through still water, it
was found that the power of the velocity to which the friction is propor-
tional varies for different surfaces, being sometimes less than 2 and some-
times more.2 Also that for long boards / (chap. ii. art. 9) is much less
than for short ones, the reason being that the forward pjirt of a long board
communicates motion to the water, and the succeeding portion thus experi-
ences less resistance.
' For results of some recent experiments on cylinders with square and
pointed ends see Min. Proc. Inst. G.E., vol. cxvii.
2 The powers arc as follows, the boards being 50 feet long : varnish 1 -SS, tinfoil
1'83, calico 1-87, fine sand 2'06, medium sand 2'00 Tinfoil is the smoothest
Biirfuce and mediiiiii Band the rmghest. These figures do not help much in
arriving at pnutioal foiniulaj for flow,
APPENDIX A— Units
Metres and Feet. — To convert a formula based on the metre into
one based on the foot — •
For metres, V=Cr„,&S'' .... ■ W
For feet,. 3^2809 F= Cy.(3-2809i?)*S* . . . {F)
Dividing i?'by if, 3-2809 = ^^-(3^2809)». Or -^^- = (3-2809)*= I'Sll.
Similarly, \i Q = KJEi
(3-2809)3() = /f;^3-2809/(3-2809//)J.
(3-2809)8= /^/3-2809(3-2809)i.
A™,
^^=(3-2809)4 = 1-811.
C,
If in either formula the index is m instead of A, the ratio — t or
K.
t- is (3-2809)1-™. This furnishes yet another instance of the
advantage of the simple indices.
Gallons and Cubic Feet. — 1 cubic foot per minute = 6-25 gallons
per minute = 375 gallons per hour = 9000 gallons per day.
APPENDIX B — Calculation of m and n
(Chap. iv. arts. 5 and 8)
The follo"wlng is a specimen of the method of calculating : —
(1)
(2)
(3)
(4)
(5)
(6)
(V)
(8)
(9)
Height
of
Weir.
Head.
M
(ob-
served).
jir "'
m
(as-
sumed).
m
<.G+H)i
^^'"(gTh)^
jnor
col>.7-=-
col. 4.
n
'^\G+H)-i
Metres.
1-135
•75
•50
Metres.
-15
Do.
Do.
-4284
-4316
•4359
■00258
■00518
-0100
•4260
1-0056
10130
r0228
-0056
■0130
-0228
2-18
2-50
2-28
1^45
1^67
1-52
The value assumed for m is constant as long as the con-
traction is complete, and it then increases according to the rules
of art. 3.
343
344
HYDRAULICS
There is a certain margin within which m may vary. The
following statement shows the values of n, calculated as above and
corresponding to different values of lU, for all the five weirs used
by Bazin and for four different heads : —
Heiglit
of Weir.
Head.
M as
observed.
Three assumed sets of values for m, and for each the
corresponding value of ?i.
(1)
(2)
(3)
w
(6)
(6)
Feet.
3-72
2-46
1-64
1-15
•79
Feet.
•49
Do.
Do.
Do.
Do.
4284
4316
4359
4424
4522
m
4250
Do.
Do.
4273
4303
n
1-45
1^67
1-52
129
•89
m
4270
Do.
Do.
4283
4313
n
■S7
1-36
137
119
•86
m i •»
4284 1 ...
Do. 1
Do. 114
4297 1 08
4327 -Si
Mean.
Do.
1-36
112
•86
3-72
2-46
1-64
1-15
■79
1-31
Do.
Do.
Do.
Do.
4286
4430
4585
4794
5034
4185
4207
4245
4305
4.395
128
1^42
120
104
•86
4200
4221
4280
4320
4410
r09
130
115
102
•85
4286
4308 ^85
4346 96
4406 -87
4500 ^75
Mean.
Do.
M6
ro8
•fi7
3-31 >
2-46
1^44
Do.
4310
4452
4167
4178
r32
161
4200
4233
102
1^28
4214 i 1 j
4275 1-07 i
Mean.
3-312
Do.
1^47
1-51
115
104
1-80 ■
4334
4100
4190 i 98
4211 •SO
1 Length of weir reduced to 3-28 feet.
2 Lengtli ofweir reduced to 1 "64 feet.
It will be noticed that slight changes in m cause great changes
in n. Obviously m cannot rise to the values shown in column 6,
as it would then equal M for the highest weirs. If reduced much
below the value of column 4 it would make h very high. The
values of in and n which seem mo.sfc suitable are those of column 5,
the mean value of n being l^l.
APPENDIX C— PouMUi.'E
Flow in Pipes (chap. v. art. 11). — The formula for flow in pipes
is sometimes put in the form , = ■ In tlus h is the head lost
L '2<j J)
in the luiifj;th L, and / is a ' friction factor ' which is equal to -^
•
APPENDIX 64:0
It is not the same as the / in equation 13, p. 21. Neither is it a
' co-efficient of friction ' which depends only on the roughness of the
surface and the velocity of the water relatively to it. It is a
variable factor which increases as C decreases. When / is '020
G is 113, and when/is -035 G is 86.
Mow in Open Channels (chap. vi. art. 11). — Houk states that at
first glance Barnes' formulse seem to agree well with experiments,
but that the observations chosen are hardly representative of the
available data and that, of the particular series chosen, only selected
measurements were included in the comparison. These contain
' such gaugings as Dubuat's ' and some in which S was determined
by aneroid barometer.
APPENDIX D— Variable Flow
(Chap. vii. art. 5)
The Ganges Canal had falls like that shown in Fig. 125 (p. 250).
Scour occurred upstream of the falls, and weirs were built on the
crests. In the JUncyclopmdia Britannica (art. Hydromechanics) it
is implied that the construction of a weir on the crest of the fall
would necessarily give a curved surface upstream. If built to the
correct height it would give the straight line BG.
APPENDIX E— Unsteady Flow
(Chap. ix. art. 1 .) Let the water from a tank be discharged over
a weir. When the water level oscillates — as when there are waves
— the discharge over the weir is slightly greater than that given by
the mean head.
(Chap. ii. art. 5, foot-note to page 332.) The bridge had three
spans of about 20 feet each. When the water in the centre bay
rose — the rise was about 6 inches — that in the side bays fell, the
fall being some 3 inches. Twenty feet upstream and down-
stream of the bridge no oscillation was perceptible from the bank.
The piers were of brickwork with acute angles at both ends, wing
walls curved. The whole period of oscillation was about twenty
seconds. The water was perhaps 6 feet deep. Possibly a small
fallen tree was submerged in the centre bay, and its branches,
pressed down by the stream, sprung back at intervals, but there
was no surface disturbance.
ODEX
HYDRAULICS IN GENERAL
Air pressure, 6, 11.
Authora.i 7, 8, 47, 70, 83, 186.
Beresford, 340.
Co-efBoients, 41.
Complex conditions, 30.
Contraction, 4.
Currents, radiating and circular,
334.
Definitions, 1, 2, 9.
Dynamic pressure, 334, 341.
Eddy, 2, 324.
Errors in observations, 41.
Expansion, 4.
Experiments (s«e Authors).
Fluid friction, 20, 15.5, 342.
Eormulae, 30, 152, 343, 344.
Gravity, 6.
History, 7.
Hydraulics, 1, 8, 9.
Impact, 337.
Irregularity of motion, 3, 299.
Module, 28.
Reaction, 336.
Rejections of observations, 296, 344.
Units, 6, 343.
Water, condition and temperature, 29,
47, 128.
Weights and measures, 5.
OBSERVATIONS
Cunningham, 305, 306.
Current meters, 291, 308.
, varieties, 310.
, rating, 315.
Discharges, 292.
Errors, 41.
'Floats, 290, 301, 324.
Flumes, 257, 293.
Gaskell, 322.
Gauges, 297, 324.
Gibson, 322.
Gourley and Crimp, 298.
Groat, 313.
Harlaoher, 311.
Judd, 323.
Observations in general, 39, 290.
Orifices, 296.
Parker, 294.
Piezometers or pressure columns, 10,
297, 299.
Pipes, 295, 821,' 322.
Pitot tubes, 294, 318.
I^ressure instruments, 291, 318, 321.
tubes, 10, 299.
Rejections of observations, 296, 334.
Rod floats, 290, 306.
Samuelson, 313.
Soundings, 256.
Stearns, 316.
Sub-surface floats, 291, 303.
Surface slope, 294, 300.
Venturi meter, 141, 295, 321.
Ward, 297.
Water-levels, 297.
Weirs, 300.
Williams, HubbellandFenkell, 295, 319.
OPEN CHANNELS
Abrupt changes, 5, 23, 31.
Backwater function, 273, 276.
Bends, 26, 83, 249, 285.
, dynamic pressure, 841.
Benton, 195.
Best form, 175.
BifurcatiouB, 88, 203, 204, 263.
variable flow, 258, 2:s, 260, 261
Bilton, 296.
Breaks in. uniformity, 254.
Canals, 87, 174, 208, 204.
Co-cffioients, 21, 144, 147, 190.
, central or surface velocity, 183,
186, 189.
, rugosity, 194, 196.
, tables, 209, 222.
' For recent authors see under tlieir respective names.
INDEX
347
OPEN CRANl^'KLS— continued
Constant velocity, 17B.
discharge, 28, 179.
Cunningham, 182, 188.
Currall, 200.
Discharge, 172.
, variable flow, 260, 280, 282.
Diversions, 262.
Equally discharging, 173.
Examples, 206, 286.
Flood waves, 328, 331.
Flumes, 257, 293.
Formulse, 20, 152, 344.
, variable flow, 109, 264, 283, 285.
Gibson, 269.
Houk, 192, 285, 344.
Jameson, 285.
Junctions, 33, 253.
, variable flow, 258, 259.
King, 204.
Manning, 192, 222.
Marr, 187.
Maximum velocity point, 180.
Momentum, 183, 284,331, 333.
Obstructions, 251.
Open channels in general, 19, 25, 27,
172.
, variable flow, 249, 345.
, unsteady flow, 328, 331, 332, 345.
Parker, 187.
Profile walls, 263.
Relative velocities in cross section, 26,
179, 254, 333.
Rivers, 197, 202, 203, 284.
Samuelson, 187, 200.
Sections, circular, 179, 248.
, irregular or special, 177, 205.
, ordinary, 173, 223, 248.
, oval, 176, 245.
Set of stream, 253.
Sewers, 176, 200.
Silt and scour, 36, 39, 106, 197, 202.
, variable flow, 250, 251, 259, 260,
261, 262.
, unsteady flow, 333.
Standing wave, 111, 266.
Surface curve, 254, 265, 271, 272, 282,
285, 288.
Thrupp, 199.
Uniform flow, 20, 24, 172.
Variable channel, 281.
• flow, 21, 22, 24, 35, 84, 109, 249.
Velocity curves, horizontal, 181, 206.
, vertical, 184, 205.
of approach (bifurcations), 203,
253.
Waves, 38, 332.
OEIFICES
Baffles, 81.
Barnes, 53.
Bell-mouthed, 12, 13, 43, 46, 61, 70.
Bilton, 53, 77.
Bovey, 53.
Condition of waiter, 47.
Co-efficients, 43, 49, 68.
, tables, 76-80.
Contraction, 18, 44, 45, 51.
Convergent, 43, 61.
Cylindrical, 43, 56, 57, 59, 77.
Divergent, 43, 64.
Drowned {see Submerged).
Examples, 72.
Farmer, 53.
Formula, 13, 48, 68, 322.
Gates {see SluicesJ.
Head, 44, 50.
Jet, 50.
Judd and King, 53, 55.
Margin, 12, 18.
Miners' inch, 56.
Nozzles, 62.
Orifices in general, 12, 16, 27, 43,
107.
Pumping, 58, 65.
Shoots, 69.
Sluices, 69.
Small heads, 15, 16, 70, 80.
Stewart, 55,
Strickland, 53.
Submerged, 14, 29, 55, 56, 79, 136.
Thin wall, 12, 13, 43, 46, 53, 55, 72.
Unsteady flow, 326.
Velocities for various heads, 75.
Velocity of approach, 13, 48, 49, 76.
PIPES
Abrupt changes, 6, 23, 31, 33, 34, 136,
155.
Air in pipes, 128, 154, 299.
Archer, 155.
Barnes, 144, 152.
Bends, 26, 33, 133, 136, 341.
Bifurcations, 33, 130.
Bilton, 135.
Brightmore, 183.
Co-efacieuts, 21, 142, 143, 148.
348
HYDRAULICS
PIPES — continued
Co-efRoients, tablcB, 160, 161, 162.
Combinations, 130.
Critical velocity, 29.
Uavis, 133.
Diaphragms, 136.
Examples, 155.
Ilamant, 144.
Formulse, 20, 152.
Garrett, 150.
Gradual changes, 139.
Hersohel, 149.
Junctions, 33.
Lawford, 144.
Mallett, 144.
Manning, 147.
Pipes in general, 19, 25, 27, 127.
Relative velocities in cross section, 26,
185.
Saph and Schoder, 144, 152.
Schoder, 133.
Schoder and Gehring, 147, 150, 152.
Scobey, 151.
Short pipes, 130.
Tables, system of, 156, 160.
, A and H, 158.
, CJS. 163.
, S, 167.
, F, 168.
Variable flow, 22, 24, 139.
Varma, 130.
Water, condition and temperature, 29,
128.
Williams, 144.
Williams and Hazen, 144.
Williams, Hubbell and Fenkell, 133.
WEIES
Air, access of, 18, 92, 97, 98.
Baffles, 81.
Canal notches, 112, 258.
Cippoletti, 93, 96.
Circular, 116.
Clear fall, 99.
Co-efflcients, 15, 81, 84, 121.
, tables, 121-127.
Contracted channels, 105.
Contraction, 18, 85, 88.
Cornell University, 97.
Crest, depth on, 85, 110.
Drowned {see Submerged).
Examples, 118.
Minn and Dyer, 93.
Flow of approach, 85.
Formulse, 15, 84, 87, 162.
Gaskell, 92.
Gourley, 116.
Gourley and Crimp, 90, 92, 93.
Harvey, 116.
Herschel, 100.
Horton, 97, 126.
Houk, 106.
Hughes, 105.
Inclined, 117.
Margin, 12, 18.
Oblique, 116, 261.
Parker, 126.
Rafter, 91.
Rapids, 98, 110, 111, 126, 267.
Rounded, 12, 82, 98, 125.
Separating, 83.
Silt and scour, 260, 261, 262.
Sloping faces or backs, 82, 97, 125,
126.
Small heads, 16, 81.
Stewart and Longwell, 90, 117.
Submerged, 16, 35, 99, 103, 105.
Thin wall, 12, 15, 82, 90, 92, 96, 110.
Trapezoidal, 93, 112, 258.
Triangular, 92, 93, 96, 111.
Values of fi|, 120.
Velocity of approach, 15, 17, 86, 121.
Water, condition of, 81.
Weir-like conditions, 105.
Weirs in general, 12, 16, 27, 81, 254,
260.
Wide crests, 82, 96, 128, 124.
Wisconsin Univei-sity, 91.
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